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{{infobox code | |||
| name = Expander codes | |||
| image = [[File:Tanner graph example.PNG|300px]] | |||
| image_caption = bipartite expander graph | |||
| namesake = | |||
| type = [[Linear block code]] | |||
| block_length = <math>n</math> | |||
| message_length = <math>n-m</math> | |||
| rate = <math>1-m/n</math> | |||
| distance = <math>2(1-\epsilon)\gamma\cdot n</math> | |||
| alphabet_size = <math>2</math> | |||
| notation = <math>[n,n-m,2(1-\epsilon)\gamma\cdot n]_2</math>-code | |||
}} | |||
In [[coding theory]], '''expander codes''' form a class of [[Error detection and correction|error-correcting codes]] that are constructed from [[Bipartite graph|bipartite]] [[expander graph]]s. | |||
Along with [[Justesen code]]s, expander codes are of particular interest since they have a constant positive [[Block_code#The_rate_R|rate]], a constant positive relative [[Block_code#The_distance_d|distance]], and a constant [[Block_code#The_alphabet_.CE.A3|alphabet size]]. | |||
In fact, the alphabet contains only two elements, so expander codes belong to the class of [[binary code]]s. | |||
Furthermore, expander codes can be both encoded and decoded in time proportional to the block length of the code. | |||
Expander codes are the only known asymptotically good codes which can be both encoded and decoded from a constant fraction of errors in polynomial time. | |||
==Expander codes== | |||
In [[coding theory]], an expander code is a <math>[n,n-m]_2\,</math> [[linear block code]] whose parity check matrix is the adjacency matrix of a bipartite [[expander graph]]. These codes have good relative [[Block_code#The_distance_d|distance]] <math>2(1-\varepsilon)\gamma\,</math>, where <math>\varepsilon\,</math> and <math>\gamma\,</math> are properties of the expander graph as defined later), [[Block_code#The_rate_R|rate]] <math>\left(1-\tfrac{m}{n}\right)\,</math>, and decodability (algorithms of running time <math>O(n)\,</math> exist). | |||
==Definition== | |||
Consider a [[bipartite graph]] <math>G(L,R,E)\,</math>, where <math>L\,</math> and <math>R\,</math> are the vertex sets and <math>E\,</math> is the set of edges connecting vertices in <math>L\,</math> to vertices of <math>R\,</math>. Suppose every vertex in <math>L\,</math> has [[degree (graph theory)|degree]] <math>d\,</math> (the graph is <math>d\,</math>-[[Regular graph|regular]]), <math>|L|=n\,</math>, and <math>|R|=m\,</math>, <math>m < n\,</math>. Then <math>G\,</math> is a <math>(N, M, d, \gamma, \alpha)\,</math> expander graph if every small enough subset <math>S \subset L\,</math>, <math>|S| \leq \gamma n\,</math> has the property that <math>S\,</math> has at least <math>d\alpha|S|\,</math> distinct neighbors in <math>R\,</math>. Note that this holds trivially for <math>\gamma \leq \tfrac{1}{n}\,</math>. When <math>\tfrac{1}{n} < \gamma \leq 1\,</math> and <math>\alpha = 1 - \varepsilon\,</math> for a constant <math>\varepsilon\,</math>, we say that <math>G\,</math> is a lossless expander. | |||
Since <math>G\,</math> is a bipartite graph, we may consider its <math>n \times m\,</math> adjacency matrix. Then the linear code <math>C\,</math> generated by viewing the transpose of this matrix as a parity check matrix is an expander code. | |||
It has been shown that nontrivial lossless expander graphs exist. Moreover, we can explicitly construct them.<ref name="lossless">{{cite book |first1=M. |last1=Capalbo |first2=O. |last2=Reingold |first3=S. |last3=Vadhan |first4=A. |last4=Wigderson |chapter=Randomness conductors and constant-degree lossless expanders |chapterurl=http://dl.acm.org/citation.cfm?id=510003 |editor= |title=STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing |publisher=ACM |location= |year=2002 |isbn=1-58113-495-9 |pages=659–668 |url= |doi=10.1145/509907.510003}}</ref> | |||
==Rate== | |||
The rate of <math>C\,</math> is its dimension divided by its block length. In this case, the parity check matrix has size <math>m \times n\,</math>, and hence <math>C\,</math> has dimension at least <math>(n-m)/n = 1 - \tfrac{m}{n}\,</math>. | |||
==Distance== | |||
Suppose <math>\varepsilon < \tfrac{1}{2}\,</math>. Then the distance of a <math>(n, m, d, \gamma, 1-\varepsilon)\,</math> expander code <math>C\,</math> is at least <math>2(1-\varepsilon)\gamma n\,</math>. | |||
===Proof=== | |||
Note that we can consider every codeword <math>c\,</math> in <math>C\,</math> as a subset of vertices <math>S \subset L\,</math>, by saying that vertex <math>v_i \in S\,</math> if and only if the <math>i\,</math>th index of the codeword is a 1. Then <math>c\,</math> is a codeword iff every vertex <math>v \in R\,</math> is adjacent to an even number of vertices in <math>S\,</math>. (In order to be a codeword, <math>cP = 0\,</math>, where <math>P\,</math> is the parity check matrix. Then, each vertex in <math>R\,</math> corresponds to each column of <math>P\,</math>. Matrix multiplication over <math>\text{GF}(2) = \{0,1\}\,</math> then gives the desired result.) So, if a vertex <math>v \in R\,</math> is adjacent to a single vertex in <math>S\,</math>, we know immediately that <math>c\,</math> is not a codeword. Let <math>N(S)\,</math> denote the neighbors in <math>R\,</math> of <math>S\,</math>, and <math>U(S)\,</math> denote those neighbors of <math>S\,</math> which are unique, i.e., adjacent to a single vertex of <math>S\,</math>. | |||
==== Lemma 1 ==== | |||
For every <math>S \subset L\,</math> of size <math>|S| \leq \gamma n\,</math>, <math>d|S| \geq |N(S)| \geq |U(S)| \geq d(1-2\varepsilon)|S|\,</math>. | |||
==== Proof ==== | |||
Trivially, <math>|N(S)| \geq |U(S)|\,</math>, since <math>v \in U(S)\,</math> implies <math>v \in N(S)\,</math>. <math>|N(S)| \leq d|S|\,</math> follows since the degree of every vertex in <math>S\,</math> is <math>d\,</math>. By the expansion property of the graph, there must be a set of <math>d(1-\varepsilon)|S|\,</math> edges which go to distinct vertices. The remaining <math>d\varepsilon|S|\,</math> edges make at most <math>d\varepsilon|S|\,</math> neighbors not unique, so <math>U(S) \geq d(1-\varepsilon)|S| - d\varepsilon|S| = d(1-2\varepsilon)|S|\,</math>. | |||
==== Corollary ==== | |||
Every sufficiently small <math>S\,</math> has a unique neighbor. This follows since <math>\varepsilon < \tfrac{1}{2}\,</math>. | |||
==== Lemma 2 ==== | |||
Every subset <math>T \subset L\,</math> with <math>|T| < 2(1-\varepsilon)\gamma n\,</math> has a unique neighbor. | |||
==== Proof ==== | |||
Lemma 1 proves the case <math>|T| \leq \gamma n\,</math>, so suppose <math>2(1-\varepsilon)\gamma n > |T| > \gamma n\,</math>. Let <math>S \subset T\,</math> such that <math>|S| = \gamma n\,</math>. By Lemma 1, we know that <math>|U(S)| \geq d(1-2\varepsilon)|S|\,</math>. Then a vertex <math>v \in U(S)\,</math> is in <math>U(T)\,</math> iff <math>v \notin N(T \setminus S)\,</math>, and we know that <math>|T \setminus S| \leq 2(1-\varepsilon)\gamma n - \gamma n = (1-2\varepsilon)\gamma n\,</math>, so by the first part of Lemma 1, we know <math>|N(T \setminus S)| \leq d(1-2\varepsilon)\gamma n\,</math>. Since <math>\varepsilon < \tfrac{1}{2}\,</math>, <math>|U(T)| \geq |U(S) \setminus N(T \setminus S)| \geq |U(S)| - |N(T \setminus S)| > 0\,</math>, and hence <math>U(T)\,</math> is not empty. | |||
==== Corollary ==== | |||
Note that if a <math>T \subset L\,</math> has at least 1 unique neighbor, i.e. <math>|U(T)| > 0\,</math>, then the corresponding word <math>c\,</math> corresponding to <math>T\,</math> cannot be a codeword, as it will not multiply to the all zeros vector by the parity check matrix. By the previous argument, <math>c \in C \implies wt(c) \geq 2(1-\varepsilon)\gamma n\,</math>. Since <math>C\,</math> is linear, we conclude that <math>C\,</math> has distance at least <math>2(1-\varepsilon)\gamma n\,</math>. | |||
==Encoding== | |||
The encoding time for an expander code is upper bounded by that of a general linear code - <math>O(n^2)\,</math> by matrix multiplication. A result due to Spielman shows that encoding is possible in <math>O(n)\,</math> time.<ref name="spielman">{{cite journal |first=D. |last=Spielman |title=Linear-time encodable and decodable error-correcting codes |journal=IEEE Transactions on Information Theory |volume=42 |issue=6 |pages=1723–31 |year=1996 |doi=10.1109/18.556668 |url=}}</ref> | |||
==Decoding== | |||
Decoding of expander codes is possible in <math>O(n)\,</math> time when <math>\varepsilon < \tfrac{1}{4}\,</math> using the following algorithm. | |||
Let <math>v_i\,</math> be the vertex of <math>L\,</math> that corresponds to the <math>i\,</math>th index in the codewords of <math>C\,</math>. Let <math>y \in \{0,1\}^n\,</math> be a received word, and <math>V(y) = \{v_i | \text{ the } i^{\text{th}} \text{ position of } y \text{ is a } 1\}\,</math>. Let <math>e(i)\,</math> be <math>|\{v \in R | N(v) \cap V(y)\,</math> is even<math>\}|\,</math>, and <math>o(i)\,</math> be <math>|\{v \in R | N(v) \cap V(y) \,</math> is odd<math>\}|\,</math>. Then consider the greedy algorithm: | |||
---- | |||
'''Input:''' received codeword <math>y\,</math>. | |||
<code> | |||
initialize y' to y | |||
while there is a v in R adjacent to an odd number of vertices in V(y') | |||
if there is an i such that o(i) > e(i) | |||
flip entry i in y' | |||
else | |||
fail</code> | |||
'''Output:''' fail, or modified codeword <math>y'\,</math>. | |||
---- | |||
=== Proof === | |||
We show first the correctness of the algorithm, and then examine its running time. | |||
==== Correctness ==== | |||
We must show that the algorithm terminates with the correct codeword when the received codeword is within half the code's distance of the original codeword. Let the set of corrupt variables be <math>S\,</math>, <math>s = |S|\,</math>, and the set of unsatisfied (adjacent to an odd number of vertices) vertices in <math>R\,</math> be <math>c\,</math>. The following lemma will prove useful. | |||
===== Lemma 3 ===== | |||
If <math>0 < s < \gamma n\,</math>, then there is a <math>v_i\,</math> with <math>o(i) > e(i)\,</math>. | |||
===== Proof ===== | |||
By Lemma 1, we know that <math>U(S) \geq d(1-2\varepsilon)s\,</math>. So an average vertex has at least <math>d(1-2\varepsilon) > d/2\,</math> unique neighbors (recall unique neighbors are unsatisfied and hence contribute to <math>o(i)\,</math>), since <math>\varepsilon < \tfrac{1}{4}\,</math>, and thus there is a vertex <math>v_i\,</math> with <math>o(i) > e(i)\,</math>. | |||
So, if we have not yet reached a codeword, then there will always be some vertex to flip. Next, we show that the number of errors can never increase beyond <math>\gamma n\,</math>. | |||
===== Lemma 4 ===== | |||
If we start with <math>s < \gamma(1-2\varepsilon)n\,</math>, then we never reach <math>s = \gamma n\,</math> at any point in the algorithm. | |||
===== Proof ===== | |||
When we flip a vertex <math>v_i\,</math>, <math>o(i)\,</math> and <math>e(i)\,</math> are interchanged, and since we had <math>o(i) > e(i)\,</math>, this means the number of unsatisfied vertices on the right decreases by at least one after each flip. Since <math>s < \gamma(1-2\varepsilon)n\,</math>, the initial number of unsatisfied vertices is at most <math>d\gamma(1-2\varepsilon)n\,</math>, by the graph's <math>d\,</math>-regularity. If we reached a string with <math>\gamma n\,</math> errors, then by Lemma 1, there would be at least <math>d\gamma(1-2\varepsilon)n\,</math> unique neighbors, which means there would be at least <math>d\gamma(1-2\varepsilon)n\,</math> unsatisfied vertices, a contradiction. | |||
Lemmas 3 and 4 show us that if we start with <math>s < \gamma(1-2\varepsilon)n\,</math> (half the distance of <math>C\,</math>), then we will always find a vertex <math>v_i\,</math> to flip. Each flip reduces the number of unsatisfied vertices in <math>R\,</math> by at least 1, and hence the algorithm terminates in at most <math>m\,</math> steps, and it terminates at some codeword, by Lemma 3. (Were it not at a codeword, there would be some vertex to flip). Lemma 4 shows us that we can never be farther than <math>\gamma n\,</math> away from the correct codeword. Since the code has distance <math>2(1-\varepsilon)\gamma n > \gamma n\,</math> (since <math>\varepsilon < \tfrac{1}{2}\,</math>), the codeword it terminates on must be the correct codeword, since the number of bit flips is less than half the distance (so we couldn't have traveled far enough to reach any other codeword). | |||
==== Complexity ==== | |||
We now show that the algorithm can achieve linear time decoding. Let <math>\tfrac{n}{m}\,</math> be constant, and <math>r\,</math> be the maximum degree of any vertex in <math>R\,</math>. Note that <math>r\,</math> is also constant for known constructions. | |||
# Pre-processing: It takes <math>O(mr)\,</math> time to compute whether each vertex in <math>R\,</math> has an odd or even number of neighbors. | |||
# Pre-processing 2: We take <math>O(dn) = O(dmr)\,</math> time to compute a list of vertices <math>v_i\,</math> in <math>L\,</math> which have <math>o(i) > e(i)\,</math>. | |||
# Each Iteration: We simply remove the first list element. To update the list of odd / even vertices in <math>R\,</math>, we need only update <math>O(d)\,</math> entries, inserting / removing as necessary. We then update <math>O(dr)\,</math> entries in the list of vertices in <math>L\,</math> with more odd than even neighbors, inserting / removing as necessary. Thus each iteration takes <math>O(dr)\,</math> time. | |||
# As argued above, the total number of iterations is at most <math>m\,</math>. | |||
This gives a total runtime of <math>O(mdr) = O(n)\,</math> time, where <math>d\,</math> and <math>r\,</math> are constants. | |||
==See also== | |||
* [[Expander graph]] | |||
* [[Low-density parity-check code]] | |||
* Linear time encoding and decoding of error-correcting codes | |||
* ABNNR and AEL codes | |||
==Notes== | |||
This article is based on Dr. Venkatesan Guruswami's course notes.<ref>{{cite web |first=V. |last=Guruswami |title=Lecture 13: Expander Codes |date=15 November 2006 |work=CSE 533: Error-Correcting |publisher=University of Washington |url=http://www.cs.washington.edu/education/courses/cse533/06au/lecnotes/lecture13.pdf |format=PDF}}<br/> | |||
{{cite web |first=V. |last=Guruswami |title=Notes 8: Expander Codes and their decoding |date=March 2010 |work=Introduction to Coding Theory |publisher=Carnegie Mellon University |url=http://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes8.pdf |format=PDF}}<br/> | |||
{{cite journal |first=V. |last=Guruswami |title=Guest column: error-correcting codes and expander graphs |journal=ACM SIGACT News |volume=35 |issue=3 |pages=25–41 |date=September 2004 |doi=10.1145/1027914.1027924 |url=http://dl.acm.org/citation.cfm?id=1027924}}</ref> | |||
==References== | |||
<references /> | |||
[[Category:Error detection and correction]] | |||
[[Category:Coding theory]] | |||
[[Category:Capacity-approaching codes]] |
Latest revision as of 23:05, 14 October 2012
Template:Cleanup Template:Infobox code
In coding theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs. Along with Justesen codes, expander codes are of particular interest since they have a constant positive rate, a constant positive relative distance, and a constant alphabet size. In fact, the alphabet contains only two elements, so expander codes belong to the class of binary codes. Furthermore, expander codes can be both encoded and decoded in time proportional to the block length of the code. Expander codes are the only known asymptotically good codes which can be both encoded and decoded from a constant fraction of errors in polynomial time.
Expander codes
In coding theory, an expander code is a linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph. These codes have good relative distance , where and are properties of the expander graph as defined later), rate , and decodability (algorithms of running time exist).
Definition
Consider a bipartite graph , where and are the vertex sets and is the set of edges connecting vertices in to vertices of . Suppose every vertex in has degree (the graph is -regular), , and , . Then is a expander graph if every small enough subset , has the property that has at least distinct neighbors in . Note that this holds trivially for . When and for a constant , we say that is a lossless expander.
Since is a bipartite graph, we may consider its adjacency matrix. Then the linear code generated by viewing the transpose of this matrix as a parity check matrix is an expander code.
It has been shown that nontrivial lossless expander graphs exist. Moreover, we can explicitly construct them.[1]
Rate
The rate of is its dimension divided by its block length. In this case, the parity check matrix has size , and hence has dimension at least .
Distance
Suppose . Then the distance of a expander code is at least .
Proof
Note that we can consider every codeword in as a subset of vertices , by saying that vertex if and only if the th index of the codeword is a 1. Then is a codeword iff every vertex is adjacent to an even number of vertices in . (In order to be a codeword, , where is the parity check matrix. Then, each vertex in corresponds to each column of . Matrix multiplication over then gives the desired result.) So, if a vertex is adjacent to a single vertex in , we know immediately that is not a codeword. Let denote the neighbors in of , and denote those neighbors of which are unique, i.e., adjacent to a single vertex of .
Lemma 1
Proof
Trivially, , since implies . follows since the degree of every vertex in is . By the expansion property of the graph, there must be a set of edges which go to distinct vertices. The remaining edges make at most neighbors not unique, so .
Corollary
Every sufficiently small has a unique neighbor. This follows since .
Lemma 2
Every subset with has a unique neighbor.
Proof
Lemma 1 proves the case , so suppose . Let such that . By Lemma 1, we know that . Then a vertex is in iff , and we know that , so by the first part of Lemma 1, we know . Since , , and hence is not empty.
Corollary
Note that if a has at least 1 unique neighbor, i.e. , then the corresponding word corresponding to cannot be a codeword, as it will not multiply to the all zeros vector by the parity check matrix. By the previous argument, . Since is linear, we conclude that has distance at least .
Encoding
The encoding time for an expander code is upper bounded by that of a general linear code - by matrix multiplication. A result due to Spielman shows that encoding is possible in time.[2]
Decoding
Decoding of expander codes is possible in time when using the following algorithm.
Let be the vertex of that corresponds to the th index in the codewords of . Let be a received word, and . Let be is even, and be is odd. Then consider the greedy algorithm:
initialize y' to y
while there is a v in R adjacent to an odd number of vertices in V(y')
if there is an i such that o(i) > e(i)
flip entry i in y'
else
fail
Output: fail, or modified codeword .
Proof
We show first the correctness of the algorithm, and then examine its running time.
Correctness
We must show that the algorithm terminates with the correct codeword when the received codeword is within half the code's distance of the original codeword. Let the set of corrupt variables be , , and the set of unsatisfied (adjacent to an odd number of vertices) vertices in be . The following lemma will prove useful.
Lemma 3
Proof
By Lemma 1, we know that . So an average vertex has at least unique neighbors (recall unique neighbors are unsatisfied and hence contribute to ), since , and thus there is a vertex with .
So, if we have not yet reached a codeword, then there will always be some vertex to flip. Next, we show that the number of errors can never increase beyond .
Lemma 4
If we start with , then we never reach at any point in the algorithm.
Proof
When we flip a vertex , and are interchanged, and since we had , this means the number of unsatisfied vertices on the right decreases by at least one after each flip. Since , the initial number of unsatisfied vertices is at most , by the graph's -regularity. If we reached a string with errors, then by Lemma 1, there would be at least unique neighbors, which means there would be at least unsatisfied vertices, a contradiction.
Lemmas 3 and 4 show us that if we start with (half the distance of ), then we will always find a vertex to flip. Each flip reduces the number of unsatisfied vertices in by at least 1, and hence the algorithm terminates in at most steps, and it terminates at some codeword, by Lemma 3. (Were it not at a codeword, there would be some vertex to flip). Lemma 4 shows us that we can never be farther than away from the correct codeword. Since the code has distance (since ), the codeword it terminates on must be the correct codeword, since the number of bit flips is less than half the distance (so we couldn't have traveled far enough to reach any other codeword).
Complexity
We now show that the algorithm can achieve linear time decoding. Let be constant, and be the maximum degree of any vertex in . Note that is also constant for known constructions.
- Pre-processing: It takes time to compute whether each vertex in has an odd or even number of neighbors.
- Pre-processing 2: We take time to compute a list of vertices in which have .
- Each Iteration: We simply remove the first list element. To update the list of odd / even vertices in , we need only update entries, inserting / removing as necessary. We then update entries in the list of vertices in with more odd than even neighbors, inserting / removing as necessary. Thus each iteration takes time.
- As argued above, the total number of iterations is at most .
This gives a total runtime of time, where and are constants.
See also
- Expander graph
- Low-density parity-check code
- Linear time encoding and decoding of error-correcting codes
- ABNNR and AEL codes
Notes
This article is based on Dr. Venkatesan Guruswami's course notes.[3]
References
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One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang