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 be a binary linear code length. 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

Basic properties

MacWilliams identity

Denote the dual code of by
.
The MacWilliams identity states that
The identity is named after Jessie MacWilliams.

Distance enumerator

The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers
where i ranges from 0 to n. The distance enumerator polynomial is
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 GFn and columns indexed by integers 0...n, and entries
The sum of the rows of B is M times the inner distribution vector.
A code C is regular if the rows of B corresponding to the codewords of C are all equal.