Enumerator polynomial

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

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

where

<math> A_t = \#\{c \in C \mid w(c) = t \} <math>

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

<math>w(c) = t<math>.

Basic properties

1. <math> W(C;0,1) = A_{0}=1 <math>
2. <math> W(C;1,1) = \sum_{w=0}^{n}A_{w}=|C| <math>
3. <math> W(C;1,0) = A_{n}= 1 \mbox{ iff } (1,\ldots,1)\in C\ \mbox{ and } 0 \mbox{ otherwise.} <math>
4. <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>

McWilliams identity

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

<math>C^\perp = \{x \in \mathbb{F}_2^n \mbox{ } \mid \mbox{ } = 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 McWilliams identity states that

<math>W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x) <math>