Enumerator polynomial

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

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

A_{t}=\#\{c\in C\mid 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

$W(C;0,1)=A_{0}=1$
$W(C;1,1)=\sum _{w=0}^{n}A_{w}=|C|$
$W(C;1,0)=A_{n}=1{\mbox{ if }}(1,\ldots ,1)\in C\ {\mbox{ and }}0{\mbox{ otherwise}}$
$W(C;1,-1)=\sum _{w=0}^{n}A_{w}(-1)^{n-w}=A_{n}+(-1)^{1}A_{n-1}+\ldots +(-1)^{n-1}A_{1}+(-1)^{n}A_{0}$

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 $\langle \ ,\ \rangle$ 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).

The identity is named after Jessie MacWilliams.

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}}\#\left\lbrace (c_{1},c_{2})\in C\times C\mid d(c_{1},c_{2})=i\right\rbrace

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}=\#\left\lbrace c\in C\mid d(c,x)=i\right\rbrace .

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.