Normal-form game: Difference between revisions

Revision as of 22:01, 12 December 2013

In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight.

Let $C \subset 𝔽_{2}^{n}$ be a binary linear code length $n$ . The weight distribution is the sequence of numbers

A_{t} = # {c \in C ∣ w (c) = t}

giving the number of codewords c in C having weight t as t ranges from 0 to n. The weight enumerator is the bivariate polynomial

W (C; x, y) = \sum_{w = 0}^{n} A_{w} x^{w} y^{n - w} .

Basic properties

MacWilliams identity

Denote the dual code of $C \subset 𝔽_{2}^{n}$ by

C^{⊥} = {x \in 𝔽_{2}^{n} ∣ ⟨ x, c ⟩ = 0 \forall c \in C}

(where $<, >$ denotes the vector dot product and which is taken over $𝔽_{2}$ ).

The MacWilliams identity states that

W (C^{⊥}; x, y) = \frac{1}{∣ C ∣} W (C; y - x, y + x) .

The identity is named after Jessie MacWilliams.

Distance enumerator

The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers

A_{i} = \frac{1}{M} # {(c_{1}, c_{2}) \in C \times C ∣ d (c_{1}, c_{2}) = i}

where i ranges from 0 to n. The distance enumerator polynomial is

A (C; x, y) = \sum_{i = 0}^{n} A_{i} x^{i} y^{n - i}

and when C is linear this is equal to the weight enumerator.

The outer distribution of C is the 2ⁿ-by-n+1 matrix B with rows indexed by elements of GF(2)ⁿ and columns indexed by integers 0...n, and entries

B_{x, i} = # {c \in C ∣ d (c, x) = i} .

The sum of the rows of B is M times the inner distribution vector (A₀,...,A_n).

A code C is regular if the rows of B corresponding to the codewords of C are all equal.

References

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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapters 3.5 and 4.3.

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+In [[coding theory]], the '''weight enumerator polynomial''' of a binary [[linear code]] specifies the number of words of each possible  [[Hamming weight]].
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+Let <math>C \subset \mathbb{F}_2^n</math> be a binary linear code length <math>n</math>.  The '''weight distribution''' is the sequence of numbers
+:<math> A_t = \#\{c \in C \mid w(c) = t \} </math>
+giving the number of [[codeword]]s ''c'' in ''C'' having weight ''t'' as ''t'' ranges from 0 to ''n''.  The '''weight enumerator''' is the bivariate [[polynomial]]
+:<math> W(C;x,y) = \sum_{w=0}^n A_w x^w y^{n-w}.</math>
+==Basic properties==
+#<math> W(C;0,1) = A_{0}=1 </math>
+#<math> W(C;1,1) = \sum_{w=0}^{n}A_{w}=|C| </math>
+#<math> W(C;1,0) = A_{n}= 1 \mbox{ iff } (1,\ldots,1)\in C\ \mbox{ and } 0 \mbox{ otherwise.} </math>
+#<math> W(C;1,-1) = \sum_{w=0}^{n}A_{w}(-1)^{n-w} = A_{n}+(-1)^{1}A_{n-1}+\ldots+(-1)^{n-1}A_{1}+(-1)^{n}A_{0} </math>
+==MacWilliams identity==
+Denote the [[dual code]] of <math>C \subset \mathbb{F}_2^n</math> by
+:<math>C^\perp = \{x \in \mathbb{F}_2^n \,\mid\, \langle x,c\rangle = 0 \mbox{  }\forall c \in C \} </math>
+(where <math><,></math> denotes the vector [[dot product]] and which is taken over <math>\mathbb{F}_2</math>).
+The '''MacWilliams identity''' states that
+:<math>W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x). </math>
+The identity is named after [[Jessie MacWilliams]].
+==Distance enumerator==
+The '''distance distribution''' or '''inner distribution''' of a code ''C'' of size ''M'' and length ''n'' is the sequence of numbers
+:<math> A_i = \frac{1}{M} \# \left\lbrace (c_1,c_2) \in C \times C \mid d(c_1,c_2) = i \right\rbrace </math>
+where ''i'' ranges from 0 to ''n''.  The '''distance enumerator polynomial''' is
+:<math> A(C;x,y) = \sum_{i=0}^n A_i x^i y^{n-i} </math>
+and when ''C'' is linear this is equal to the weight enumerator.
+The '''outer distribution''' of ''C'' is the 2<sup>''n''</sup>-by-''n''+1 matrix ''B'' with rows indexed by elements of GF(2)<sup>''n''</sup> and columns indexed by integers 0...''n'', and entries
+:<math> B_{x,i} = \# \left\lbrace c \in C \mid d(c,x) = i \right\rbrace . </math>
+The sum of the rows of ''B'' is ''M'' times the inner distribution vector (''A''<sub>0</sub>,...,''A''<sub>''n''</sub>).
+A code ''C'' is '''regular''' if the rows of ''B'' corresponding to the codewords of ''C'' are all equal.
+==References==
+* {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | publisher=[[Oxford University Press]] | series=Oxford Applied Mathematics and Computing Science Series | date=1986 | isbn=0-19-853803-0 | pages=165–173 }}
+* {{cite book | last = Pless | first = Vera | authorlink=Vera Pless | title = Introduction to the theory of error-correcting codes | publisher = [[John Wiley & Sons]]|series = Wiley-Interscience Series in Discrete Mathematics | date = 1982| isbn = 0-471-08684-3 | pages=103–119 }}
+* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd ed | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 }} Chapters 3.5 and 4.3.
+[[Category:Coding theory]]
+[[Category:Error detection and correction]]
+[[Category:Mathematical identities]]

Normal-form game: Difference between revisions

Revision as of 22:01, 12 December 2013

Contents

Basic properties

MacWilliams identity

Distance enumerator

References

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Normal-form game: Difference between revisions

Revision as of 22:01, 12 December 2013

Basic properties

MacWilliams identity

Distance enumerator

References

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