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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]].

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{ if } (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>\langle\ ,\ \rangle</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''n''-by-''n''+1 matrix ''B'' with rows indexed by elements of GF(2)''n'' 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''0,...,''A''''n'').

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

==Source==

[http://wikipedia.org/ http://wikipedia.org/]
==See Also on BitcoinWiki==
* [[Snapup]]
* [[Starrie]]
* [[IziBits]]
* [[Nametoken]]
* [[SherCoin]]

Enumerator polynomial - Revision history

Admin: Created page with "In coding theory, the '''weight enumerator polynomial''' of a binary linear code specifies the number of words of each possible Hamming weight. Let C \subse..."

Admin: Created page with "In coding theory, the '''weight enumerator polynomial''' of a binary linear code specifies the number of words of each possible Hamming weight. Let $C \subse..."$