Enumerator polynomial

In mathematics, the weight enumerator of a binary linear code $C \subset \mathbb{F}_2^n$ of length $n$ is defined to be

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

where

$A_t = \#\{c \in C \mid w(c) = t \}$

is defined to be the number of codewords c in C having Hamming weight

w (c) = t

[edit] Basic properties

$W (C;0,1) = A 0 = 1$
$W(C;1,1) = \sum_{w=0}^{n}A_{w}=|C|$
$W(C;1,0) = A_{n}= 1 \mbox{ iff } (1,\ldots,1)\in C\ \mbox{ and } 0 \mbox{ otherwise.}$
$W(C;1,-1) = \sum_{w=0}^{n}A_{w}(-1)^{n-w} = A_{n}+(-1)^{1}A_{n-1}+\ldots+(-1)^{n-1}A_{1}+(-1)^{n}A_{0}$

Denote the dual code of $C \subset \mathbb{F}_2^n$ by

$C^\perp = \{x \in \mathbb{F}_2^n \,\mid\, \langle x,c\rangle = 0 \mbox{ }\forall c \in C \}$

(where $< , >$ denotes the vector dot product and which is taken over $\mathbb{F}_2$ ).

The MacWilliams identity states that

$W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x).$