MacWilliams identities

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 length ${\displaystyle n}$. The weight distribution is the sequence of numbers

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

Denote the dual code of ${\displaystyle C\subset \mathbb {F} _{2}^{n}}$ by

...
Wikipedia