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= | In [[coding theory]], the '''BCH codes''' form a class of [[cyclic code|cyclic]] [[Error detection and correction|error-correcting codes]] that are constructed using [[finite fields]]. BCH codes were invented in 1959 by French mathematician [[Alexis Hocquenghem]], and independently in 1960 by [[Raj Chandra Bose|Raj Bose]] and [[D.K. Ray-Chaudhuri|D. K. Ray-Chaudhuri]].<ref>{{Harvnb|Reed|Chen|1999|p=189}}</ref> The acronym ''BCH'' comprises the initials of these inventors' names. | ||
One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an [[Abstract algebra|algebraic]] method known as [[syndrome decoding]]. This simplifies the design of the decoder for these codes, using small low-power electronic hardware. | |||
== | BCH codes are used in applications such as satellite communications,<ref>{{cite web|title=Phobos Lander Coding System: Software and Analysis|url=http://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF|accessdate=25 February 2012}}</ref> [[compact disc]] players, [[DVD]]s, [[Disk storage|disk drives]], [[solid-state drive]]s<ref>{{cite web|title=Sandforce SF-2500/2600 Product Brief|url=http://www.sandforce.com/index.php?id=133&parentId=2&top=1|accessdate=25 February 2012}}</ref> and [[Bar codes|two-dimensional bar codes]]. | ||
== Definition and illustration == | |||
== | === Primitive narrow-sense BCH codes === | ||
Given a prime {{mvar|q}} and positive integers {{mvar|m}} and {{mvar|d}} with {{math|''d'' ≤ ''q''<sup>''m''</sup> - 1}}, a primitive narrow-sense BCH code over the finite field {{math|GF(''q'')}} with code length {{math|''n'' {{=}} ''q''<sup>''m''</sup> - 1}} and [[Block code#The distance d|distance]] at least {{mvar|d}} is constructed by the following method. | |||
Let {{mvar|α}} be a [[Primitive element (finite field)|primitive element]] of {{math|GF(''q''<sup>''m''</sup>)}}. | |||
For any positive integer {{mvar|i}}, let {{math|''m''<sub>''i''</sub>(''x'')}} be the [[minimal polynomial (field theory)|minimal polynomial]] of {{math|α<sup>''i''</sup>}}. | |||
The generator polynomial of the BCH code is defined as the [[least common multiple]] {{math|''g''(''x'') {{=}} lcm(''m''<sub>1</sub>(''x''),…,''m''<sub>''d'' - 1</sub>(''x''))}}. | |||
It can be seen that {{math|''g''(''x'')}} is a polynomial with coefficients in {{math|GF(''q'')}} and divides {{math|''x''<sup>''n''</sup> - 1}}. | |||
Therefore, the polynomial code defined by {{math|''g''(''x'')}} is a cyclic code. | |||
== | ==== Example ==== | ||
Let {{math|''q''{{=}}2}} and {{math|''m''{{=}}4}} (therefore {{math|''n''{{=}}15}}). We will consider different values of {{mvar|d}}. There is a primitive root {{math|α}} in {{math|GF(16)}} satisfying | |||
:{{NumBlk|:|<math>\alpha^4+\alpha+1=0</math>|{{EquationRef|1}}}} | |||
its minimal polynomial over {{math|GF(2)}} is | |||
:<math>m_1(x) = x^4+x+1.</math> | |||
The minimal polynomials of the first seven powers of {{math|α}} are | |||
:<math>m_1(x) = m_2(x) = m_4(x) = x^4+x+1,\,</math> | |||
:<math>m_3(x) = m_6(x) = x^4+x^3+x^2+x+1,\,</math> | |||
:<math>m_5(x) = x^2+x+1,\,</math> | |||
:<math>m_7(x) = x^4+x^3+1.\,</math> | |||
The BCH code with <math>d=2,3</math> has generator polynomial | |||
== | <math>g(x) = m_1(x) = x^4+x+1.\,</math> | ||
It has minimal Hamming distance at least 3 and corrects up to one error. Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits. | |||
= | The BCH code with <math>d=4,5</math> has generator polynomial | ||
<math>g(x) = {\rm lcm}(m_1(x),m_3(x)) = (x^4+x+1)(x^4+x^3+x^2+x+1) = x^8+x^7+x^6+x^4+1.\,</math> | |||
It has minimal Hamming distance at least 5 and corrects up to two errors. Since the generator polynomial is of degree 8, this code has 7 data bits and 8 checksum bits. | |||
The BCH code with <math>d=8</math> and higher has generator polynomial | |||
== | <math> | ||
\begin{align} | |||
g(x) & {} = {\rm lcm}(m_1(x),m_3(x),m_5(x),m_7(x)) \\ | |||
& {} = (x^4+x+1)(x^4+x^3+x^2+x+1)(x^2+x+1)(x^4+x^3+1) \\ | |||
& {} = x^{14}+x^{13}+x^{12}+\cdots+x^2+x+1. | |||
\end{align} | |||
</math> | |||
This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. In fact, this code has only two codewords: 000000000000000 and 111111111111111. | |||
== | === General BCH codes === | ||
General BCH codes differ from primitive narrow-sense BCH codes in two respects. | |||
First, the requirement that <math>\alpha</math> be a primitive element of <math>\mathrm{GF}(q^m)</math> can be relaxed. By relaxing this requirement, the code length changes from <math>q^m - 1</math> to <math>\mathrm{ord}(\alpha),</math> the [[Order (group theory)|order]] of the element <math>\alpha.</math> | |||
Second, the consecutive roots of the generator polynomial may run from <math>\alpha^c,\ldots,\alpha^{c+d-2}</math> instead of <math>\alpha,\ldots,\alpha^{d-1}.</math> | |||
'''Definition.''' Fix a finite field <math>GF(q),</math> where <math>q</math> is a prime power. Choose positive integers <math>m,n,d,c</math> such that <math>2\leq d\leq n,</math> <math>{\rm gcd}(n,q)=1,</math> and <math>m</math> is the [[multiplicative order]] of <math>q</math> modulo <math>n.</math> | |||
= | As before, let <math>\alpha</math> be a [[primitive nth root of unity|primitive <math>n</math>th root of unity]] in <math>GF(q^m),</math> and let <math>m_i(x)</math> be the [[minimal polynomial (field theory)|minimal polynomial]] over <math>GF(q)</math> of <math>\alpha^i</math> for all <math>i.</math> | ||
The generator polynomial of the BCH code is defined as the [[least common multiple]] <math>g(x) = {\rm lcm}(m_c(x),\ldots,m_{c+d-2}(x)).</math> | |||
'''Note:''' if <math>n=q^m-1</math> as in the simplified definition, then <math>{\rm gcd}(n,q)</math> is automatically 1, and the order of <math>q</math> modulo <math>n</math> is automatically <math>m.</math> | |||
Therefore, the simplified definition is indeed a special case of the general one. | |||
== | === Special cases === | ||
* A BCH code with <math>c=1</math> is called a ''narrow-sense BCH code''. | |||
* A BCH code with <math>n=q^m-1</math> is called ''primitive''. | |||
The generator polynomial <math>g(x)</math> of a BCH code has coefficients from <math>\mathrm{GF}(q).</math> | |||
In general, a cyclic code over <math>\mathrm{GF}(q^p)</math> with <math>g(x)</math> as the generator polynomial is called a BCH code over <math>\mathrm{GF}(q^p).</math> | |||
The BCH code over <math>\mathrm{GF}(q^m)</math> with <math>g(x)</math> as the generator polynomial is called a [[Reed-Solomon]] code. In other words, a Reed-Solomon code is a BCH code where the decoder alphabet is the same as the channel alphabet.<ref>{{Harvnb|Gill|unknown|p=3}}</ref> | |||
== | == Properties == | ||
1. The generator polynomial of a BCH code has degree at most <math>(d-1)m.</math> | |||
Moreover, if <math>q=2</math> and <math>c=1,</math> the generator polynomial has degree at most <math>dm/2.</math> | |||
:Proof: each minimal polynomial <math>m_i(x)</math> has degree at most <math>m.</math> | |||
Therefore, the least common multiple of <math>d-1</math> of them has degree at most <math>(d-1)m.</math> | |||
Moreover, if <math>q=2,</math> then <math>m_i(x) = m_{2i}(x)</math> for all <math>i.</math> | |||
Therefore, <math>g(x)</math> is the least common multiple of at most <math>d/2</math> minimal polynomials <math>m_i(x)</math> for odd indices <math>i,</math> each of degree at most <math>m.</math> | |||
2. A BCH code has minimal Hamming distance at least <math>d.</math> | |||
Proof: Suppose that <math>p(x)</math> is a code word with fewer than <math>d</math> non-zero terms. Then | |||
:<math>p(x) = b_1x^{k_1} + \cdots + b_{d-1}x^{k_{d-1}},\text{ where }k_1<k_2<\cdots<k_{d-1}.</math> | |||
== | Recall that <math>\alpha^c,\ldots,\alpha^{c+d-2}</math> are roots of <math>g(x),</math> hence of <math>p(x).</math> | ||
This implies that <math>b_1,\ldots,b_{d-1}</math> satisfy the following equations, for <math>i=c,\ldots,c+d-2:</math> | |||
:<math>p(\alpha^i) = b_1\alpha^{ik_1} + b_2\alpha^{ik_2} + \cdots + b_{d-1}\alpha^{ik_{d-1}} = 0.</math> | |||
1) | In matrix form, we have | ||
:<math>\begin{bmatrix} | |||
\alpha^{ck_1} & \alpha^{ck_2} & \cdots & \alpha^{ck_{d-1}} \\ | |||
\alpha^{(c+1)k_1} & \alpha^{(c+1)k_2} & \cdots & \alpha^{(c+1)k_{d-1}} \\ | |||
\vdots & \vdots && \vdots \\ | |||
\alpha^{(c+d-2)k_1} & \alpha^{(c+d-2)k_2} & \cdots & \alpha^{(c+d-2)k_{d-1}} \\ | |||
\end{bmatrix} | |||
\begin{bmatrix} | |||
b_1 \\ b_2 \\ \vdots \\ b_{d-1} | |||
\end{bmatrix} | |||
= | |||
\begin{bmatrix} | |||
0 \\ 0 \\ \vdots \\ 0 | |||
\end{bmatrix}. | |||
</math> | |||
== | The determinant of this matrix equals | ||
:<math>\left(\prod_{i=1}^{d-1}\alpha^{ck_i}\right)\det\begin{pmatrix} | |||
1 & 1 & \cdots & 1 \\ | |||
\alpha^{k_1} & \alpha^{k_2} & \cdots & \alpha^{k_{d-1}} \\ | |||
\vdots & \vdots && \vdots \\ | |||
\alpha^{(d-2)k_1} & \alpha^{(d-2)k_2} & \cdots & \alpha^{(d-2)k_{d-1}} \\ | |||
\end{pmatrix} = \left(\prod_{i=1}^{d-1}\alpha^{ck_i}\right) \det(V).</math> | |||
The matrix <math>V</math> is seen to be a [[Vandermonde matrix]], and its determinant is | |||
:<math>\det(V) = \prod_{1\le i<j\le d-1} (\alpha^{k_j}-\alpha^{k_i}),</math> | |||
which is non-zero. It therefore follows that <math>b_1,\ldots,b_{d-1}=0,</math> hence <math>p(x)=0.</math> | |||
3. A BCH code is cyclic. | |||
Proof: A polynomial code of length <math>n</math> is cyclic if and only if its generator polynomial divides <math>x^n-1.</math> | |||
Since <math>g(x)</math> is the minimal polynomial with roots <math>\alpha^c,\ldots,\alpha^{c+d-2},</math> it suffices to check that each of <math>\alpha^c,\ldots,\alpha^{c+d-2}</math> is a root of <math>x^n-1.</math> | |||
This follows immediately from the fact that <math>\alpha</math> is, by definition, an <math>n</math>th root of unity. | |||
== Encoding == | |||
{{Empty section|date=March 2013}} | |||
== Decoding == | |||
There are many algorithms for decoding BCH codes. The most common ones follow this general outline: | |||
# Calculate the syndromes ''s<sub>j</sub>'' for the received vector <!-- there are d syndromes --> | |||
# Determine the number of errors ''t'' and the error locator polynomial ''Λ(x)'' from the syndromes <!-- Gill uses ν for actual errors; some references use ''t'' as max number of correctable errors. --> | |||
# Calculate the roots of the error location polynomial to find the error locations ''X<sub>i</sub>'' | |||
# Calculate the error values ''Y<sub>i</sub>'' at those error locations <!-- also known as e_i --> | |||
# Correct the errors | |||
During some of these steps, the decoding algorithm may determine that the received vector has too many errors and cannot be corrected. For example, if an appropriate value of ''t'' is not found, then the correction would fail. In a truncated (not primitive) code, an error location may be out of range. If the received vector has more errors than the code can correct, the decoder may unknowingly produce an apparently valid message that is not the one that was sent. | |||
===Calculate the syndromes=== | |||
The received vector <math>R</math> is the sum of the correct codeword <math>C</math> and an unknown error vector <math>E.</math> | |||
The syndrome values are formed by considering <math>R</math> as a polynomial and evaluating it at <math>\alpha^c,\ldots,\alpha^{c+d-2}.</math> | |||
Thus the syndromes are<ref>{{Harvnb|Lidl|Pilz|1999|p=229}}</ref> | |||
:<math>s_j = R(\alpha^{j}) = C(\alpha^{j}) + E(\alpha^{j})</math> | |||
for <math>j = c</math> to <math>c+d-2.</math> | |||
Since <math>\alpha^{j}</math> are the zeros of <math>g(x),</math> of which | |||
<math>C(x)</math> is a multiple, <math>C(\alpha^{j}) = 0.</math> | |||
Examining the syndrome values thus isolates the error vector so one can begin to solve for it. | |||
If there is no error, <math>s_j = 0</math> for all <math>j.</math> | |||
If the syndromes are all zero, then the decoding is done. | |||
===Calculate the error location polynomial=== | |||
If there are nonzero syndromes, then there are errors. The decoder needs to figure out how many errors and the location of those errors. | |||
If there is a single error, write this as <math>E(x) = e\,x^i,</math> | |||
where <math>i</math> is the location of the error and <math>e</math> is its magnitude. Then the first two syndromes are | |||
:<math>s_c = e\,\alpha^{c\,i}</math> | |||
:<math>s_{c+1} = e\,\alpha^{(c+1)\,i} = \alpha^i s_c</math> | |||
so together they allow us to calculate <math>e</math> and provide some information about <math>i</math> (completely determining it in the case of Reed-Solomon codes). | |||
If there are two or more errors, | |||
:<math>E(x) = e_1 x^{i_1} + e_2 x^{i_2} + \cdots \, </math> | |||
It is not immediately obvious how to begin solving the resulting syndromes for the unknowns <math>e_k</math> and <math>i_k.</math> | |||
First step is finding locator polynomial | |||
:<math>\Lambda(x)=\prod_{j=1}^t (x\alpha^{i_j}-1)</math> compatible with computed syndroms and with minimal possible <math>t.</math> | |||
Two popular algorithms for this task are: | |||
# [[BCH code#Peterson–Gorenstein–Zierler algorithm|Peterson–Gorenstein–Zierler algorithm]] | |||
# [[Berlekamp–Massey algorithm]] | |||
====Peterson–Gorenstein–Zierler algorithm==== | |||
<!-- this confuses t (max number of errors that can be corrected) with ν (actual number of errors) --> | |||
Peterson's algorithm is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients <math> \lambda_1 , \lambda_2, \dots, \lambda_{v} </math> of a polynomial | |||
: <math> \Lambda(x) = 1 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_v x^v .</math> | |||
Now the procedure of the Peterson–Gorenstein–Zierler algorithm.<ref>{{Harvnb|Gorenstein|Peterson|Zierler|1960}}</ref> Expect we have at least 2''t'' syndromes ''s<sub>c</sub>'',...,''s<sub>c+2t-1</sub>. | |||
Let ''v''=''t''. | |||
* Start by generating the <math>S_{v\times v}</math> matrix with elements that are syndrome values | |||
::<math>S_{v \times v}=\begin{bmatrix}s_c&s_{c+1}&\dots&s_{c+v-1}\\ | |||
s_{c+1}&s_{c+2}&\dots&s_{c+v}\\ | |||
\vdots&\vdots&\ddots&\vdots\\ | |||
s_{c+v-1}&s_{c+v}&\dots&s_{c+2v-2}\end{bmatrix}. | |||
</math> | |||
* Generate a <math>c_{v \times 1}</math> vector with elements | |||
::<math>C_{v \times 1}=\begin{bmatrix}s_{c+v}\\ | |||
s_{c+v+1}\\ | |||
\vdots\\ | |||
s_{c+2v-1}\end{bmatrix}. | |||
</math> | |||
* Let <math>\Lambda</math> denote the unknown polynomial coefficients, which are given by | |||
::<math>\Lambda_{v \times 1} = \begin{bmatrix}\lambda_{v}\\ | |||
\lambda_{v-1}\\ | |||
\vdots\\ | |||
\lambda_{1}\end{bmatrix}. | |||
</math> | |||
* Form the matrix equation | |||
::<math>S_{v \times v} \Lambda_{v \times 1} = -C_{v \times 1\,} .</math> | |||
* If the determinant of matrix <math>S_{v \times v}</math> is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown <math>\Lambda</math> values. | |||
* If <math> \det(S_{v \times v}) = 0 ,</math> then follow | |||
if <math>v = 0</math> | |||
then | |||
declare an empty error locator polynomial | |||
stop Peterson procedure. | |||
end | |||
set <math> v \leftarrow v -1</math> | |||
continue from the beginning of Peterson's decoding by making smaller <math>S_{v \times v}</math> | |||
* After you have values of <math>\Lambda</math>, you have with you the error locator polynomial. | |||
* Stop Peterson procedure. | |||
===Factor error locator polynomial=== | |||
Now that you have the <math>\Lambda(x)</math> polynomial, its roots can be found in the form <math>\Lambda(x)=(\alpha^{i_1} x - 1) (\alpha^{i_2} x - 1) \cdots (\alpha^{i_v} x - 1)</math> by brute force for example using the [[Chien search]] algorithm. The exponential | |||
powers of the primitive element <math>\alpha</math> will yield the positions where errors occur in the received word; hence the name 'error locator' polynomial. | |||
The zeros of ''Λ(x)'' are ''α<sup>-i<sub>1</sub></sup>'', ..., ''α<sup>-i<sub>v</sub></sup>''. | |||
===Calculate error values=== | |||
Once the error locations are known, the next step is to determine the error values at those locations. The error values are then used to correct the received values at those locations to recover the original codeword. | |||
For the case of binary BCH, (with all characters readable) this is trivial; just flip the bits for the received word at these positions, and we have the corrected code word. In the more general case, the error weights <math>e_j</math> can be determined by solving the linear system | |||
:<math>s_c = e_1 \alpha^{c\,i_1} + e_2 \alpha^{c\,i_2} + \ldots</math> | |||
:<math>s_{c+1} = e_1 \alpha^{(c + 1)\,i_1} + e_2 \alpha^{(c + 1)\,i_2} + \ldots</math> | |||
:. . . | |||
==== Forney algorithm ==== | |||
However, there is a more efficient method known as the [[Forney algorithm]]. | |||
Let <math>S(x)=s_c+s_{c+1}x+s_{c+2}x^2+\cdots+s_{c+d-2}x^{d-2}.</math> | |||
Let <math>v\le d-1,</math> <math>\lambda_0\neq 0,</math> and <math>\Lambda(x)=\sum_{i=0}^v\lambda_ix^i=\lambda_0\cdot\prod_{k=0}^{v} (\alpha^{-i_k}x-1).</math> | |||
Let <math>\Omega(x) = S(x)\,\Lambda(x) \pmod{x^{d-1}}</math> be the error evaluator polynomial<ref name="Gill-Forney">{{Harvnb|Gill|unknown|p=47}}</ref> | |||
Let <math>\Lambda'(x) = \Sigma_{i=1}^v i \cdot \lambda_i x^{i-1},</math> where <math>i\cdot x</math> denotes here <math>\textstyle\sum_{k=1}^i x</math> rather than multiplying in the field. | |||
Than if syndromes could be explained by an error word, which could be nonzero only on positions <math>i_k</math>, then error values are | |||
:<math>e_k=-{\alpha^{i_k}\Omega(\alpha^{-i_k})\over \alpha^{c\cdot i_k}\Lambda'(\alpha^{-i_k})}.</math> | |||
For narrow-sense BCH codes, ''c'' = 1, so the expression simplifies to: | |||
:<math>e_k=-{\Omega(\alpha^{-i_k})\over \Lambda'(\alpha^{-i_k})}.</math> | |||
==== Explanation of Forney algorithm computation ==== | |||
It is based on [[Lagrange polynomial|Lagrange interpolation]] and techniques of [[generating function]]s. | |||
Look at <math>S(x)\Lambda(x).</math> | |||
Let for simplicity <math>\lambda_k=0</math> for <math>k>v,</math> and <math>s_k=0</math> for <math>k>c+d-2.</math> | |||
Then <math>S(x)\Lambda(x)=\sum_{j=0}^{\infty}\sum_{i=0}^j s_{j-i+1}\lambda_i x^j.</math> | |||
:<math>S(x)=\sum_{i=0}^{d-2}\sum_{j=1}^v e_j\alpha^{(c+i)\cdot i_j} x^i=\sum_{j=1}^v e_j\alpha^{c\,i_j}\sum_{i=0}^{d-2}(\alpha^{i_j})^i x^i = \sum_{j=1}^v e_j\alpha^{c\,i_j} {(x\alpha^{i_j})^{d-1}-1\over x\alpha^{i_j}-1}.</math> | |||
:<math>S(x)\Lambda(x)=S(x)\lambda_0\prod_{\ell=1}^v (\alpha^{i_\ell}x-1)=\lambda_0\sum_{j=1}^v e_j\alpha^{c\,i_j} {(x\alpha^{i_j})^{d-1}-1\over x\alpha^{i_j}-1}\prod_{\ell=1}^v (\alpha^{i_\ell}x-1).</math> | |||
We could gain form of polynomial: | |||
:<math>S(x)\Lambda(x)=\lambda_0\sum_{j=1}^v e_j\alpha^{c\,i_j} ((x\alpha^{i_j})^{d-1}-1)\prod_{\ell\in\{1,\dots,v\}\setminus\{j\}} (\alpha^{i_\ell}x-1).</math> | |||
We want to compute unknowns <math>e_j,</math> and we could simplify the context by removing the <math>(x\alpha^{i_j})^{d-1}</math> terms. This leads to the error evaluator polynomial | |||
:<math>\Omega(x) = S(x)\,\Lambda(x) \pmod{x^{d-1}}.</math> | |||
Thanks to <math>v\le d-1</math> we have | |||
:<math>\Omega(x) = -\lambda_0\sum_{j=1}^v e_j\alpha^{c\,i_j} \prod_{\ell\in\{1,\dots,v\}\setminus\{j\}} (\alpha^{i_\ell}x-1).</math> | |||
Look at <math>\Omega(\alpha^{-i_k}).</math> | |||
Thanks to <math>\Lambda</math> (the Lagrange interpolation trick) the sum degenerates to only one summand | |||
:<math>\Omega(\alpha^{-i_k})=-\lambda_0e_k\alpha^{c\cdot i_k}\prod_{\ell\in\{1,\dots,v\}\setminus\{k\}}(\alpha^{i_\ell}\alpha^{-i_k}-1). | |||
</math> | |||
To get <math>e_k</math> we just should get rid of the product. We could compute the product directly from already computed roots <math>\alpha^{-i_j}</math> of <math>\Lambda,</math> but we could use simpler form. | |||
As [[formal derivative]] <math>\Lambda'(x)=\lambda_0\sum_{j=1}^v \alpha^{i_j}\prod_{\ell\in\{1,\dots,v\}\setminus\{j\}}(\alpha^{i_\ell}x-1),</math> | |||
we get again only one summand in | |||
:<math>\Lambda'(\alpha^{-i_k})= | |||
\lambda_0\alpha^{i_k}\prod_{\ell\in\{1,\dots,v\}\setminus\{k\}}(\alpha^{i_\ell}\alpha^{-i_k}-1) | |||
.</math> | |||
So finally | |||
:<math>e_k=-{\alpha^{i_k}\Omega(\alpha^{-i_k})\over \alpha^{c\cdot i_k}\Lambda'(\alpha^{-i_k})}.</math> | |||
This formula is advantageous when one computes the formal derivative of <math>\Lambda</math> form its <math>\Lambda(x)=\sum_{i=1}^v\lambda_ix^i</math> form, gaining | |||
:<math>\Lambda'(x) = \Sigma_{i=1}^v i \cdot \lambda_i x^{i-1},</math> | |||
where <math>i\cdot x</math> denotes here <math>\textstyle\sum_{k=1}^i x</math> rather than multiplying in the field. | |||
=== Decoding based on extended Euclidean algorithm === | |||
The process of finding both the polynomial Λ and the error values could be based on the [[Extended Euclidean algorithm]]. Correction of unreadable characters could be incorporated to the algorithm easily as well. | |||
Let <math>k_1, ... ,k_k</math> be positions of unreadable characters. One creates polynomial localising these positions <math>\Gamma(x)=\prod_{i=1}^k(x\alpha^{k_i}-1).</math> | |||
Set values on unreadable positions to 0 and compute the syndromes. | |||
As we have already defined for the Forney formula let <math>S(x)=\sum_{i=0}^{d-2}s_{c+i}x^i.</math> | |||
Let us run extended Euclidean algorithm for locating least common divisor of polynomials <math>S(x)\Gamma(x)</math> and <math>x^{d-1}.</math> | |||
The goal is not to find the least common divisor, but a polynomial <math>r(x)</math> of degree at most <math>\lfloor (d+k-3)/2\rfloor</math> and polynomials <math>a(x), b(x)</math> such that <math>r(x)=a(x)S(x)\Gamma(x)+b(x)x^{d-1}.</math> | |||
Low degree of <math>r(x)</math> guarantees, that <math>a(x)</math> would satisfy extended (by <math>\Gamma</math>) defining conditions for <math>\Lambda.</math> | |||
Defining <math>\Xi(x)=a(x)\Gamma(x)</math> and using <math>\Xi</math> on the place of <math>\Lambda(x)</math> in the Fourney formula will give us error values. | |||
The main advantage of the algorithm is that it meanwhile computes <math>\Omega(x)=S(x)\Xi(x)\bmod x^{d-1}=r(x)</math> required in the Forney formula. | |||
==== Explanation of the decoding process ==== | |||
The goal is to find a codeword which differs from the received word minimally as possible on readable positions. | |||
When expressing the received word as a sum of nearest codeword and error word, we are trying to find error word with minimal number of non-zeros on readable positions. | |||
Syndrom <math>s_i</math> restricts error word by condition <math>s_i=\sum_{j=0}^{n-1}e_j\alpha^{ij}.</math> | |||
We could write these conditions separately or we could create polynomial <math>S(x)=\sum_{i=0}^{d-2}s_{c+i}x^i</math> and compare coefficients near powers <math>0</math> to <math>d-2.</math> | |||
<math>S(x){\textstyle{\{0,\ldots,\,d-2\}\atop =}}E(x)=\sum_{i=0}^{d-2}\sum_{j=0}^{n-1}e_j\alpha^{ij}\alpha^{cj}x^i.</math> | |||
Suppose there is unreadable letter on position <math>k_1,</math> we could replace set of syndroms <math>\{s_c,\ldots,s_{c+d-2}\}</math> by set of syndroms <math>\{t_c,\ldots,t_{c+d-3}\}</math> defined by equation <math>t_i=\alpha^{k_1}s_i-s_{i+1}.</math> | |||
Suppose for an error word all restrictions by original set <math>\{s_c,\ldots,s_{c+d-2}\}</math> of syndroms hold, | |||
than <math>t_i=\alpha^{k_1}s_i-s_{i+1}=\alpha^{k_1}\sum_{j=0}^{n-1}e_j\alpha^{ij}-\sum_{j=0}^{n-1}e_j\alpha^j\alpha^{ij}=\sum_{j=0}^{n-1}e_j(\alpha^{k_1}-\alpha^j)\alpha^{ij}.</math> | |||
New set of syndroms restricts error vector <math>f_j=e_j(\alpha^{k_1}-\alpha^j)</math> the same way the original set of syndroms restricted the error vector <math>e_j.</math> | |||
Note, that except the coordinate <math>k_1,</math> where <math>f_{k_1}=0,</math> an <math>f_j</math> is zero, iff <math>e_j</math> is zero. | |||
For the goal of locating error positions we could change the set of syndroms in the similar way to reflect all unreadable characters. | |||
This shortens the set of syndroms by <math>k.</math> | |||
In polynomial formulation, the replacement of syndroms set <math>\{s_c,\ldots,s_{c+d-2}\}</math> by syndroms set <math>\{t_c,\ldots,t_{c+d-3}\}</math> leads to <math>T(x)=\sum_{i=0}^{d-3}t_{c+i}x^i=\alpha^{k_1}\sum_{i=0}^{d-3}s_{c+i}x^i-\sum_{i=1}^{d-2}s_{c+i}x^{i-1}.</math> | |||
Therefore <math>xT(x){\textstyle{\{1,\ldots,\,d-2\}\atop =}}(x\alpha^{k_1}-1)S(x).</math> | |||
After replacement of <math>S(x)</math> by <math>S(x)\Gamma(x)</math>, one would require equation for coefficients near powers <math>k,\ldots,d-2.</math> | |||
One could consider looking for error positions from the point of view of eliminating influence of given positions similarly as for unreadable characters. | |||
If we found <math>v</math> positions such that eliminating their influence leads to obtaining set of syndroms consisting of all zeros, | |||
than there exists error vector with errors only on these coordinates. | |||
If <math>\Lambda(x)</math> denotes the polynomial eliminating the influence of these coordinates, we obtain <math>S(x)\Gamma(x)\Lambda(x){\textstyle{\{k+v,\ldots,\,d-2\}\atop =}}0.</math> | |||
In Euclidean algorithm, we try to correct at most <math>(d-1-k)/2</math> errors (on readable positions), because with bigger error count there could be more codewords in the same distance from the received word. | |||
Therefore, for <math>\Lambda(x)</math> we are looking for, the equation must hold for coefficients near powers starting from <math>k+\lfloor(d-1-k)/2\rfloor.</math> | |||
In Forney formula, <math>\Lambda(x)</math> could be multiplied by a scalar giving the same result. | |||
It could happen that the Euclidean algorithm finds <math>\Lambda(x)</math> of degree higher than <math>(d-1-k)/2,</math> having number of different roots equal to its degree, | |||
where the Fourney formula would be able to correct errors in all its roots, anyways correcting such many errors could be risky (especially with no other restrictions on received word). | |||
Usually after getting <math>\Lambda(x)</math> of higher degree, we decide not to correct the errors. | |||
Correction could fail in the case <math>\Lambda(x)</math> has roots with higher multiplicity or the number of roots is smaller than its degree. | |||
Fail could be detected as well by Forney formula returning error outside the transmitted alphabet. | |||
===Correct the errors=== | |||
Using the error values and error location, correct the errors and form a corrected code vector by subtracting error values at error locations. | |||
===Decoding examples=== | |||
==== Decoding of binary code without unreadable characters ==== | |||
Consider a BCH code in GF(2<sup>4</sup>) with <math>d=7</math> and <math>g(x) = x^{10} + x^8 + x^5 + x^4 + x^2 + x + 1</math>. (This is used in [[QR code]]s.) Let the message to be transmitted be <nowiki>[1 1 0 1 1]</nowiki>, or in polynomial notation, <math>M(x) = x^4 + x^3 + x + 1.</math> | |||
The "checksum" symbols are calculated by dividing <math>x^{10} M(x)</math> by <math>g(x)</math> and taking the remainder, resulting in <math>x^9 + x^4 + x^2</math> or <nowiki>[ 1 0 0 0 0 1 0 1 0 0 ]</nowiki>. These are appended to the message, so the transmitted codeword is <nowiki>[ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 ]</nowiki>. | |||
Now, imagine that there are two bit-errors in the transmission, so the received codeword is [ 1 {{color|red|0}} 0 1 1 1 0 0 0 {{color|red|1}} 1 0 1 0 0 ]. In polynomial notation: | |||
:<math>R(x) = C(x) + x^{13} + x^5 = x^{14} + x^{11} + x^{10} + x^9 + x^5 + x^4 + x^2</math> | |||
In order to correct the errors, first calculate the syndromes. Taking <math>\alpha = 0010,</math> we have <math>s_1 = R(\alpha^1) = 1011,</math> <math>s_2 = 1001,</math> <math>s_3 = 1011,</math> <math>s_4 = 1101,</math> <math>s_5 = 0001,</math> and <math>s_6 = 1001.</math> | |||
Next, apply the Peterson procedure by row-reducing the following augmented matrix. | |||
:<math>\left [ S_{3 \times 3} | C_{3 \times 1} \right ] = | |||
\begin{bmatrix}s_1&s_2&s_3&s_4\\ | |||
s_2&s_3&s_4&s_5\\ | |||
s_3&s_4&s_5&s_6\end{bmatrix} = | |||
\begin{bmatrix}1011&1001&1011&1101\\ | |||
1001&1011&1101&0001\\ | |||
1011&1101&0001&1001\end{bmatrix} \Rightarrow | |||
\begin{bmatrix}0001&0000&1000&0111\\ | |||
0000&0001&1011&0001\\ | |||
0000&0000&0000&0000\end{bmatrix}</math> | |||
Due to the zero row, {{math|''S''<sub>3×3</sub>}} is singular, which is no surprise since only two errors were introduced into the codeword. | |||
However, the upper-left corner of the matrix is identical to {{closed-closed|''S''<sub>2×2</sub> <nowiki>|</nowiki> ''C''<sub>2×1</sub>}}, which gives rise to the solution <math>\lambda_2 = 1000,</math> <math>\lambda_1 = 1011.</math> | |||
The resulting error locator polynomial is <math>\Lambda(x) = 1000 x^2 + 1011 x + 0001,</math> which has zeros at <math>0100 = \alpha^{-13}</math> and <math>0111 = \alpha^{-5}.</math> | |||
The exponents of <math>\alpha</math> correspond to the error locations. | |||
There is no need to calculate the error values in this example, as the only possible value is 1. | |||
==== Decoding with unreadable characters ==== | |||
Suppose the same scenario, but the received word has two unreadable characters [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 {{color|red|1}} 1 0 1 0 0 ]. | |||
We replace the unreadable characters by zeros while creating the polynom reflecting their positions <math>\Gamma(x)=(\alpha^8x-1)(\alpha^{11}x-1).</math> | |||
We compute the syndroms <math>s_1=\alpha^{-7},</math> <math>s_2=\alpha^{1},</math> <math>s_3=\alpha^{4},</math> | |||
<math>s_4=\alpha^{2},</math> <math>s_5=\alpha^{5},</math> and <math>s_6=\alpha^{-7}.</math> | |||
(Using log notation which is independent on GF(2<sup>4</sup>) isomorphisms. | |||
For computation checking we can use the same representation for addition as was used in previous example. | |||
Hexadecimal description of the powers of <math>\alpha</math> are consecutively 1,2,4,8,3,6,C,B,5,A,7,E,F,D,9 with the addition based on bitwise xor.) | |||
Let us make syndrome polynomial <math>S(x)=\alpha^{-7}+\alpha^{1}x+\alpha^{4}x^2+\alpha^{2}x^3+\alpha^{5}x^4+\alpha^{-7}x^5,</math> | |||
compute | |||
<math>S(x)\Gamma(x)=\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5+\alpha^{7}x^6+\alpha^{-3}x^7.</math> | |||
Run Extended Euclidean algorithm: | |||
<math> | |||
\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix}= | |||
\begin{pmatrix}\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5+\alpha^{7}x^6+\alpha^{-3}x^7\\ x^6\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{7}+\alpha^{-3}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix}x^6\\ | |||
\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5+(\alpha^{7}+\alpha^{7})x^6+(\alpha^{-3}+\alpha^{-3})x^7\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{7}+\alpha^{-3}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix}\alpha^4+\alpha^{-5}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix} | |||
\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5\\ | |||
\alpha^{-3}+(\alpha^{-7}+\alpha^{3})x+(\alpha^{3}+\alpha^{-1})x^2+\\ | |||
(\alpha^{-5}+\alpha^{-6})x^3+(\alpha^3+\alpha^{1})x^4+(\alpha^{-6}+\alpha^{-6})x^5+(\alpha^0+1)x^6\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}(1+\alpha^{-4})+(\alpha^{1}+\alpha^{2})x+\alpha^{7}x^2&\alpha^{7}+\alpha^{-3}x | |||
\\ | |||
\alpha^4+\alpha^{-5}x&1\end{pmatrix} | |||
\begin{pmatrix} | |||
\alpha^{-7}+\alpha^{4}x+\alpha^{-1}x^2+ | |||
\alpha^{6}x^3+\alpha^{-1}x^4+\alpha^{5}x^5\\ | |||
\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^2+ | |||
\alpha^{-2}x^3+\alpha^{-6}x^4\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^2&\alpha^{7}+\alpha^{-3}x | |||
\\ | |||
\alpha^4+\alpha^{-5}x&1\end{pmatrix} | |||
\begin{pmatrix}\alpha^{-5}+\alpha^{-4}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix} | |||
\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^2+ | |||
\alpha^{-2}x^3+\alpha^{-6}x^4\\ | |||
(\alpha^{7}+\alpha^{-7})+ | |||
(\alpha^{-7}+\alpha^{-7}+\alpha^{4})x+\\ | |||
(\alpha^{-5}+\alpha^{-6}+\alpha^{-1})x^2+\\ | |||
(\alpha^{-7}+\alpha^{-4}+\alpha^{6})x^3+\\ | |||
(\alpha^{4}+\alpha^{-6}+\alpha^{-1})x^4+ | |||
(\alpha^{5}+\alpha^{5})x^5\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^2\\ | |||
\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2&\alpha^4+\alpha^{-5}x\end{pmatrix} | |||
\begin{pmatrix} | |||
\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^2+ | |||
\alpha^{-2}x^3+\alpha^{-6}x^4\\ | |||
\alpha^{-4}+\alpha^{4}x+\alpha^{2}x^2+ | |||
\alpha^{-5}x^3\end{pmatrix}. | |||
</math> | |||
We have reached polynomial of degree at most 3, and as | |||
<math> | |||
\begin{pmatrix}-(\alpha^4+\alpha^{-5}x)&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^2\\ | |||
\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2&-(\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3)\end{pmatrix} | |||
\begin{pmatrix}\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^2\\ | |||
\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2&\alpha^4+\alpha^{-5}x\end{pmatrix} | |||
=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}, | |||
</math> | |||
we get | |||
<math> | |||
\begin{pmatrix}-(\alpha^4+\alpha^{-5}x)&\alpha^{-3}+\alpha^{5}x+\alpha^{7}x^2\\ | |||
\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2&-(\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3)\end{pmatrix} | |||
\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix}= | |||
\begin{pmatrix} | |||
\alpha^{-3}+\alpha^{-2}x+\alpha^{0}x^2+\\ | |||
\alpha^{-2}x^3+\alpha^{-6}x^4\\ | |||
\alpha^{-4}+\alpha^{4}x+\alpha^{2}x^2+\\ | |||
\alpha^{-5}x^3\end{pmatrix}. | |||
</math> | |||
Therefore | |||
<math> | |||
S(x)\Gamma(x)(\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2)- | |||
(\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3)x^6= | |||
\alpha^{-4}+\alpha^{4}x+\alpha^{2}x^2+\alpha^{-5}x^3. | |||
</math> | |||
Let <math>\Lambda(x)=\alpha^{3}+\alpha^{-5}x+\alpha^{6}x^2.</math> | |||
Don't worry that <math>\lambda_0\neq 1.</math> | |||
Find by brute force a root of <math>\Lambda.</math> | |||
The roots are <math>\alpha^2,</math> and <math>\alpha^{10}</math> | |||
(after finding for example <math>\alpha^2</math> we can divide <math>\Lambda</math> by corresponding monom <math>(x-\alpha^2)</math> and the root of resulting monom could be found easily). | |||
Let <math>\Xi(x)=\Gamma(x)\Lambda(x)=\alpha^3+\alpha^4x^2+\alpha^2x^3+\alpha^{-5}x^4,</math> | |||
and let <math>\Omega(x)=S(x)\Xi(x)\bmod x^6=\alpha^{-4}+\alpha^4x+\alpha^2x^2+\alpha^{-5}x^3.</math> | |||
Let us look for error values using formula <math>e_j=-\Omega(\alpha^{-i_j})/\Xi'(\alpha^{-i_j}),</math> | |||
where <math>\alpha^{-i_j}</math> are roots of <math>\Xi(x).</math> | |||
<math>\Xi'(x)=\alpha^{2}x^2.</math> | |||
We get <math>e_1=-\Omega(\alpha^4)/\Xi'(\alpha^{4})=(\alpha^{-4}+\alpha^{-7}+\alpha^{-5}+\alpha^{7})/\alpha^{-5}=\alpha^{-5}/\alpha^{-5}=1,</math> | |||
<math>e_2=-\Omega(\alpha^7)/\Xi'(\alpha^{7})=(\alpha^{-4}+\alpha^{-4}+\alpha^{1}+\alpha^{1})/\alpha^{1}=0,</math> | |||
<math>e_3=-\Omega(\alpha^{10})/\Xi'(\alpha^{10})=(\alpha^{-4}+\alpha^{-1}+\alpha^{7}+\alpha^{-5})/\alpha^{7}=\alpha^{7}/\alpha^{7}=1,</math> | |||
<math>e_4=-\Omega(\alpha^{2})/\Xi'(\alpha^{2})=(\alpha^{-4}+\alpha^{6}+\alpha^{6}+\alpha^{1})/\alpha^{6}=\alpha^{6}/\alpha^{6}=1.</math> | |||
Fact, that <math>e_3=e_4=1,</math> should not be surprising. | |||
Corrected code is therefore [ 1 {{color|green|1}} 0 {{color|green|1}} 1 1 {{color|green|0}} 0 0 {{color|green|0}} 1 0 1 0 0]. | |||
==== Decoding with unreadable characters with a small number of errors ==== | |||
Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ]. | |||
Again, replace the unreadable characters by zeros while creating the polynom reflecting their positions <math>\Gamma(x)=(\alpha^8x-1)(\alpha^{11}x-1).</math> | |||
Compute the syndroms | |||
<math>s_1=\alpha^{4},</math> <math>s_2=\alpha^{-7},</math> <math>s_3=\alpha^{1},</math> | |||
<math>s_4=\alpha^{1},</math> <math>s_5=\alpha^{0},</math> and <math>s_6=\alpha^{2}.</math> | |||
Create syndrom polynommial | |||
<math>S(x)=\alpha^{4}+\alpha^{-7}x+\alpha^{1}x^2+\alpha^{1}x^3+\alpha^{0}x^4+\alpha^{2}x^5,</math> | |||
and <math>S(x)\Gamma(x)=\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5+\alpha^{-1}x^6+\alpha^{6}x^7.</math> | |||
Let us run Extended Euclidean algorithm: | |||
<math> | |||
\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix}= | |||
\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5+\alpha^{-1}x^6+\alpha^{6}x^7\\ | |||
x^6\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{-1}+\alpha^{6}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix}x^6\\ | |||
\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5+(\alpha^{-1}+\alpha^{-1})x^6+(\alpha^{6}+\alpha^{6})x^7 | |||
\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}\alpha^{-1}+\alpha^{6}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix}\alpha^{3}+\alpha^{1}x&1\\ 1&0\end{pmatrix} | |||
\begin{pmatrix} | |||
\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5\\ | |||
\alpha^{7}+(\alpha^{-5}+\alpha^{5})x+ | |||
(\alpha^{-7}+\alpha^{-7})x^2+ | |||
(\alpha^{6}+\alpha^{6})x^3+\\ | |||
(\alpha^{4}+\alpha^{4})x^4+ | |||
(\alpha^{2}+\alpha^{2})x^5+ | |||
(\alpha^{0}+1)x^6 | |||
\end{pmatrix}= | |||
</math> | |||
<math> | |||
\begin{pmatrix}(1+\alpha^{2})+(\alpha^{0}+\alpha^{-6})x+\alpha^{7}x^2&\alpha^{-1}+\alpha^{6}x\\ | |||
\alpha^{3}+\alpha^{1}x&1\end{pmatrix} | |||
\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5\\ | |||
\alpha^{7}+\alpha^{0}x | |||
\end{pmatrix}. | |||
</math> | |||
We have reached polynomial of degree at most 3, and as | |||
<math> | |||
\begin{pmatrix}-(1)&\alpha^{-1}+\alpha^{6}x\\ | |||
\alpha^{3}+\alpha^{1}x&-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^2)\end{pmatrix} | |||
\begin{pmatrix}\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^2&\alpha^{-1}+\alpha^{6}x\\ | |||
\alpha^{3}+\alpha^{1}x&1\end{pmatrix} | |||
=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}, | |||
</math> | |||
we get | |||
<math> | |||
\begin{pmatrix}-(1)&\alpha^{-1}+\alpha^{6}x\\ | |||
\alpha^{3}+\alpha^{1}x&-(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^2)\end{pmatrix} | |||
\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix}= | |||
\begin{pmatrix}\alpha^{4}+\alpha^{7}x+\alpha^{5}x^2+\alpha^{3}x^3+\alpha^{1}x^4+\alpha^{-1}x^5\\ | |||
\alpha^{7}+\alpha^{0}x | |||
\end{pmatrix}. | |||
</math> | |||
Therefore, | |||
<math> | |||
S(x)\Gamma(x)(\alpha^{3}+\alpha^{1}x)- | |||
(\alpha^{-7}+\alpha^{7}x+\alpha^{7}x^2)x^6= | |||
\alpha^{7}+\alpha^{0}x. | |||
</math> | |||
Let <math>\Lambda(x)=\alpha^{3}+\alpha^{1}x.</math> | |||
Don't worry that <math>\lambda_0\neq 1.</math> | |||
The root of <math>\Lambda(x)</math> is <math>\alpha^{3-1}.</math> | |||
Let <math>\Xi(x)=\Gamma(x)\Lambda(x)=\alpha^{3}+\alpha^{-7}x+\alpha^{-4}x^2+\alpha^{5}x^3,</math> and <math>\Omega(x)=S(x)\Xi(x)\bmod x^6=\alpha^{7}+\alpha^{0}x.</math> | |||
Let us look for error values using formula <math>e_j=-\Omega(\alpha^{-i_j})/\Xi'(\alpha^{-i_j}),</math> | |||
where <math>\alpha^{-i_j}</math> are roots of polynomial <math>\Xi(x).</math> | |||
<math>\Xi'(x)=\alpha^{-7}+\alpha^{5}x^2.</math> | |||
We get <math>e_1=-\Omega(\alpha^4)/\Xi'(\alpha^{4})=(\alpha^{7}+\alpha^{4})/(\alpha^{-7}+\alpha^{-2})=\alpha^{3}/\alpha^{3}=1,</math> | |||
<math>e_2=-\Omega(\alpha^7)/\Xi'(\alpha^{7})=(\alpha^{7}+\alpha^{7})/(\alpha^{-7}+\alpha^{4})=0/\alpha^{5}=0,</math> | |||
<math>e_3=-\Omega(\alpha^2)/\Xi'(\alpha^2)=(\alpha^{7}+\alpha^{2})/(\alpha^{-7}+\alpha^{-6})=\alpha^{-3}/\alpha^{-3}=1.</math> | |||
Fact, that <math>e_3=1,</math> should not be surprising. | |||
Corrected code is therefore [ 1 {{color|green|1}} 0 {{color|green|1}} 1 1 {{color|green|0}} 0 0 0 1 0 1 0 0]. | |||
==Citations== | |||
{{reflist|30em}} | |||
== References == | |||
===Primary sources=== | |||
* {{Citation | |||
|first= A. | |||
|last= Hocquenghem | |||
|author-link= Alexis Hocquenghem | |||
|title=Codes correcteurs d'erreurs | |||
|language= French | |||
|journal= Chiffres | |||
|location= Paris | |||
|volume=2 | |||
|pages= 147–156 | |||
|date= September 1959 | |||
|issn= | |||
|doi=}} | |||
* {{Citation | |||
|first= R. C. | |||
|last= Bose | |||
|author-link= R. C. Bose | |||
|first2= D. K. | |||
|last2= Ray-Chaudhuri | |||
|author2-link= D. K. Ray-Chaudhuri | |||
|title= On A Class of Error Correcting Binary Group Codes | |||
|journal= [[Information and Control]] | |||
|volume= 3 | |||
|issue=1 | |||
|pages= 68–79 | |||
|date= March 1960 | |||
|issn= 0890-5401 | |||
|doi=}} | |||
===Secondary sources=== | |||
* {{Citation | |||
|first= W. J. | |||
|last= Gilbert | |||
|first2= W. K. | |||
|last2= Nicholson | |||
|title= Modern Algebra with Applications | |||
|edition= 2nd | |||
|publisher= John Wiley | |||
|year= 2004 | |||
|url= | |||
|isbn= | |||
|doi=}} | |||
* {{Citation | |||
|last=Gill | |||
|first= John | |||
|title= EE387 Notes #7, Handout #28 | |||
|accessdate= April 21, 2010 | |||
|pages=42–45 | |||
|publisher= Stanford University | |||
|url= http://www.stanford.edu/class/ee387/handouts/notes7.pdf | |||
|doi=}}{{dead link|date=November 2012}} Course notes are apparently being redone for 2012: http://www.stanford.edu/class/ee387/ | |||
* {{Citation | |||
|last= Gorenstein | |||
|first= Daniel | |||
|authorlink= Daniel Gorenstein | |||
|last2= Peterson | |||
|first2= W. Wesley | |||
|authorlink2= W. Wesley Peterson | |||
|last3= Zierler | |||
|first3 = Neal | |||
|authorlink3= Neal Zierler | |||
|title= Two-Error Correcting Bose-Chaudhuri Codes are Quasi-Perfect | |||
|journal= Information and Control | |||
|volume= 3 | |||
|issue= 3 | |||
|pages= 291–294 | |||
|year= 1960 | |||
|doi= }} | |||
* {{Citation | |||
|first= Rudolf | |||
|last= Lidl | |||
|first2= Günter | |||
|last2= Pilz | |||
|title= Applied Abstract Algebra | |||
|edition= 2nd | |||
|publisher= John Wiley | |||
|year= 1999 | |||
|url= | |||
|isbn= | |||
|doi=}} | |||
* {{Citation | |||
|first= S. | |||
|last= Lin | |||
|first2= D. | |||
|last2= Costello | |||
|title= Error Control Coding: Fundamentals and Applications | |||
|publisher= Prentice-Hall | |||
|location= Englewood Cliffs, NJ | |||
|year= 2004 | |||
|isbn= | |||
|doi= }} | |||
* {{Citation | |||
|first= F. J. | |||
|last=MacWilliams | |||
|first2= N. J. A. | |||
|last2= Sloane | |||
|authorlink2= N. J. A. Sloane | |||
|title= The Theory of Error-Correcting Codes | |||
|location= New York, NY | |||
|publisher= North-Holland Publishing Company | |||
|year= 1977 | |||
|isbn= | |||
|doi=}} | |||
* {{Citation | |||
|first= Irving S. | |||
|last= Reed | |||
|authorlink= Irving S. Reed | |||
|first2= Xuemin | |||
|last2= Chen | |||
|title= Error-Control Coding for Data Networks | |||
|location= Boston, MA | |||
|publisher= [[Kluwer Academic Publishers]] | |||
|year= 1999 | |||
|isbn= 0-7923-8528-4 | |||
|doi=}} | |||
* {{Citation | |||
|first= Atri | |||
|last= Rudra | |||
|title= CSE 545, Error Correcting Codes: Combinatorics, Algorithms and Applications | |||
|publisher= University at Buffalo | |||
|url= http://www.cse.buffalo.edu/~atri/courses/coding-theory/ | |||
|accessdate= April 21, 2010 | |||
|doi=}} | |||
{{DEFAULTSORT:Bch Code}} | |||
[[Category:Error detection and correction]] | |||
[[Category:Finite fields]] | |||
[[Category:Coding theory]] | |||
Latest revision as of 05:48, 18 September 2013
In coding theory, the BCH codes form a class of cyclic error-correcting codes that are constructed using finite fields. BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Bose and D. K. Ray-Chaudhuri.[1] The acronym BCH comprises the initials of these inventors' names.
One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-power electronic hardware.
BCH codes are used in applications such as satellite communications,[2] compact disc players, DVDs, disk drives, solid-state drives[3] and two-dimensional bar codes.
Definition and illustration
Primitive narrow-sense BCH codes
Given a prime Template:Mvar and positive integers Template:Mvar and Template:Mvar with Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., a primitive narrow-sense BCH code over the finite field Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. with code length Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and distance at least Template:Mvar is constructed by the following method.
Let Template:Mvar be a primitive element of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..
For any positive integer Template:Mvar, let Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. be the minimal polynomial of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..
The generator polynomial of the BCH code is defined as the least common multiple Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..
It can be seen that Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is a polynomial with coefficients in Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and divides Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..
Therefore, the polynomial code defined by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is a cyclic code.
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Let Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. (therefore Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.). We will consider different values of Template:Mvar. There is a primitive root Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. in Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. satisfying
its minimal polynomial over Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is
The minimal polynomials of the first seven powers of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. are
The BCH code with has generator polynomial
It has minimal Hamming distance at least 3 and corrects up to one error. Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits.
The BCH code with has generator polynomial
It has minimal Hamming distance at least 5 and corrects up to two errors. Since the generator polynomial is of degree 8, this code has 7 data bits and 8 checksum bits.
The BCH code with and higher has generator polynomial
This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. In fact, this code has only two codewords: 000000000000000 and 111111111111111.
General BCH codes
General BCH codes differ from primitive narrow-sense BCH codes in two respects.
First, the requirement that be a primitive element of can be relaxed. By relaxing this requirement, the code length changes from to the order of the element
Second, the consecutive roots of the generator polynomial may run from instead of
Definition. Fix a finite field where is a prime power. Choose positive integers such that and is the multiplicative order of modulo
As before, let be a primitive th root of unity in and let be the minimal polynomial over of for all The generator polynomial of the BCH code is defined as the least common multiple
Note: if as in the simplified definition, then is automatically 1, and the order of modulo is automatically Therefore, the simplified definition is indeed a special case of the general one.
Special cases
The generator polynomial of a BCH code has coefficients from In general, a cyclic code over with as the generator polynomial is called a BCH code over The BCH code over with as the generator polynomial is called a Reed-Solomon code. In other words, a Reed-Solomon code is a BCH code where the decoder alphabet is the same as the channel alphabet.[4]
Properties
1. The generator polynomial of a BCH code has degree at most Moreover, if and the generator polynomial has degree at most
Therefore, the least common multiple of of them has degree at most Moreover, if then for all Therefore, is the least common multiple of at most minimal polynomials for odd indices each of degree at most
2. A BCH code has minimal Hamming distance at least Proof: Suppose that is a code word with fewer than non-zero terms. Then
Recall that are roots of hence of This implies that satisfy the following equations, for
In matrix form, we have
The determinant of this matrix equals
The matrix is seen to be a Vandermonde matrix, and its determinant is
which is non-zero. It therefore follows that hence
3. A BCH code is cyclic.
Proof: A polynomial code of length is cyclic if and only if its generator polynomial divides Since is the minimal polynomial with roots it suffices to check that each of is a root of This follows immediately from the fact that is, by definition, an th root of unity.
Encoding
Decoding
There are many algorithms for decoding BCH codes. The most common ones follow this general outline:
- Calculate the syndromes sj for the received vector
- Determine the number of errors t and the error locator polynomial Λ(x) from the syndromes
- Calculate the roots of the error location polynomial to find the error locations Xi
- Calculate the error values Yi at those error locations
- Correct the errors
During some of these steps, the decoding algorithm may determine that the received vector has too many errors and cannot be corrected. For example, if an appropriate value of t is not found, then the correction would fail. In a truncated (not primitive) code, an error location may be out of range. If the received vector has more errors than the code can correct, the decoder may unknowingly produce an apparently valid message that is not the one that was sent.
Calculate the syndromes
The received vector is the sum of the correct codeword and an unknown error vector The syndrome values are formed by considering as a polynomial and evaluating it at Thus the syndromes are[5]
for to Since are the zeros of of which is a multiple, Examining the syndrome values thus isolates the error vector so one can begin to solve for it.
If there is no error, for all If the syndromes are all zero, then the decoding is done.
Calculate the error location polynomial
If there are nonzero syndromes, then there are errors. The decoder needs to figure out how many errors and the location of those errors.
If there is a single error, write this as where is the location of the error and is its magnitude. Then the first two syndromes are
so together they allow us to calculate and provide some information about (completely determining it in the case of Reed-Solomon codes).
If there are two or more errors,
It is not immediately obvious how to begin solving the resulting syndromes for the unknowns and First step is finding locator polynomial
Two popular algorithms for this task are:
Peterson–Gorenstein–Zierler algorithm
Peterson's algorithm is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients of a polynomial
Now the procedure of the Peterson–Gorenstein–Zierler algorithm.[6] Expect we have at least 2t syndromes sc,...,sc+2t-1. Let v=t.
- Form the matrix equation
- If the determinant of matrix is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown values.
- If then follow
if then declare an empty error locator polynomial stop Peterson procedure. end set continue from the beginning of Peterson's decoding by making smaller
Factor error locator polynomial
Now that you have the polynomial, its roots can be found in the form by brute force for example using the Chien search algorithm. The exponential powers of the primitive element will yield the positions where errors occur in the received word; hence the name 'error locator' polynomial.
The zeros of Λ(x) are α-i1, ..., α-iv.
Calculate error values
Once the error locations are known, the next step is to determine the error values at those locations. The error values are then used to correct the received values at those locations to recover the original codeword.
For the case of binary BCH, (with all characters readable) this is trivial; just flip the bits for the received word at these positions, and we have the corrected code word. In the more general case, the error weights can be determined by solving the linear system
Forney algorithm
However, there is a more efficient method known as the Forney algorithm.
Let be the error evaluator polynomial[7]
Let where denotes here rather than multiplying in the field.
Than if syndromes could be explained by an error word, which could be nonzero only on positions , then error values are
For narrow-sense BCH codes, c = 1, so the expression simplifies to:
Explanation of Forney algorithm computation
It is based on Lagrange interpolation and techniques of generating functions.
Look at Let for simplicity for and for
We could gain form of polynomial:
We want to compute unknowns and we could simplify the context by removing the terms. This leads to the error evaluator polynomial
Look at Thanks to (the Lagrange interpolation trick) the sum degenerates to only one summand
To get we just should get rid of the product. We could compute the product directly from already computed roots of but we could use simpler form.
As formal derivative we get again only one summand in
So finally
This formula is advantageous when one computes the formal derivative of form its form, gaining
where denotes here rather than multiplying in the field.
Decoding based on extended Euclidean algorithm
The process of finding both the polynomial Λ and the error values could be based on the Extended Euclidean algorithm. Correction of unreadable characters could be incorporated to the algorithm easily as well.
Let be positions of unreadable characters. One creates polynomial localising these positions Set values on unreadable positions to 0 and compute the syndromes.
As we have already defined for the Forney formula let
Let us run extended Euclidean algorithm for locating least common divisor of polynomials and The goal is not to find the least common divisor, but a polynomial of degree at most and polynomials such that Low degree of guarantees, that would satisfy extended (by ) defining conditions for
Defining and using on the place of in the Fourney formula will give us error values.
The main advantage of the algorithm is that it meanwhile computes required in the Forney formula.
Explanation of the decoding process
The goal is to find a codeword which differs from the received word minimally as possible on readable positions. When expressing the received word as a sum of nearest codeword and error word, we are trying to find error word with minimal number of non-zeros on readable positions. Syndrom restricts error word by condition We could write these conditions separately or we could create polynomial and compare coefficients near powers to
Suppose there is unreadable letter on position we could replace set of syndroms by set of syndroms defined by equation Suppose for an error word all restrictions by original set of syndroms hold, than New set of syndroms restricts error vector the same way the original set of syndroms restricted the error vector Note, that except the coordinate where an is zero, iff is zero. For the goal of locating error positions we could change the set of syndroms in the similar way to reflect all unreadable characters. This shortens the set of syndroms by
In polynomial formulation, the replacement of syndroms set by syndroms set leads to Therefore
After replacement of by , one would require equation for coefficients near powers
One could consider looking for error positions from the point of view of eliminating influence of given positions similarly as for unreadable characters. If we found positions such that eliminating their influence leads to obtaining set of syndroms consisting of all zeros, than there exists error vector with errors only on these coordinates. If denotes the polynomial eliminating the influence of these coordinates, we obtain
In Euclidean algorithm, we try to correct at most errors (on readable positions), because with bigger error count there could be more codewords in the same distance from the received word. Therefore, for we are looking for, the equation must hold for coefficients near powers starting from
In Forney formula, could be multiplied by a scalar giving the same result.
It could happen that the Euclidean algorithm finds of degree higher than having number of different roots equal to its degree, where the Fourney formula would be able to correct errors in all its roots, anyways correcting such many errors could be risky (especially with no other restrictions on received word). Usually after getting of higher degree, we decide not to correct the errors. Correction could fail in the case has roots with higher multiplicity or the number of roots is smaller than its degree. Fail could be detected as well by Forney formula returning error outside the transmitted alphabet.
Correct the errors
Using the error values and error location, correct the errors and form a corrected code vector by subtracting error values at error locations.
Decoding examples
Decoding of binary code without unreadable characters
Consider a BCH code in GF(24) with and . (This is used in QR codes.) Let the message to be transmitted be [1 1 0 1 1], or in polynomial notation, The "checksum" symbols are calculated by dividing by and taking the remainder, resulting in or [ 1 0 0 0 0 1 0 1 0 0 ]. These are appended to the message, so the transmitted codeword is [ 1 1 0 1 1 1 0 0 0 0 1 0 1 0 0 ].
Now, imagine that there are two bit-errors in the transmission, so the received codeword is [ 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 1 1 1 0 0 0 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 1 0 1 0 0 ]. In polynomial notation:
In order to correct the errors, first calculate the syndromes. Taking we have and Next, apply the Peterson procedure by row-reducing the following augmented matrix.
Due to the zero row, Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is singular, which is no surprise since only two errors were introduced into the codeword.
However, the upper-left corner of the matrix is identical to Template:Closed-closed, which gives rise to the solution
The resulting error locator polynomial is which has zeros at and
The exponents of correspond to the error locations.
There is no need to calculate the error values in this example, as the only possible value is 1.
Decoding with unreadable characters
Suppose the same scenario, but the received word has two unreadable characters [ 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 ? 1 1 ? 0 0 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the polynom reflecting their positions We compute the syndroms and (Using log notation which is independent on GF(24) isomorphisms. For computation checking we can use the same representation for addition as was used in previous example. Hexadecimal description of the powers of are consecutively 1,2,4,8,3,6,C,B,5,A,7,E,F,D,9 with the addition based on bitwise xor.)
Let us make syndrome polynomial compute
Run Extended Euclidean algorithm:
We have reached polynomial of degree at most 3, and as we get
Let Don't worry that Find by brute force a root of The roots are and (after finding for example we can divide by corresponding monom and the root of resulting monom could be found easily).
Let and let Let us look for error values using formula where are roots of We get Fact, that should not be surprising.
Corrected code is therefore [ 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 1 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 0 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 1 0 1 0 0].
Decoding with unreadable characters with a small number of errors
Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ].
Again, replace the unreadable characters by zeros while creating the polynom reflecting their positions Compute the syndroms and Create syndrom polynommial and Let us run Extended Euclidean algorithm:
We have reached polynomial of degree at most 3, and as we get
Let Don't worry that The root of is
Let and Let us look for error values using formula where are roots of polynomial We get Fact, that should not be surprising.
Corrected code is therefore [ 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 1 1 23 yr old Mining Engineer (Excluding Oil ) Roman Kull from Markham, enjoys to spend time metal detecting, property developers in singapore property index and history. In recent time took some time to check out Macquarie Island. 0 0 0 1 0 1 0 0].
Citations
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
Primary sources
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Secondary sources
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010Template:Dead link Course notes are apparently being redone for 2012: http://www.stanford.edu/class/ee387/ - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010