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

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

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

${\displaystyle W(C;x,y)=\sum _{w=0}^{n}A_{w}x^{w}y^{n-w}.}$