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 formula_1 be a binary linear code length formula_2. 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
MacWilliams identity.
Denote the dual code of formula_1 by
(where formula_11 denotes the vector dot product and which is taken over formula_12).
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 2"n"-by-"n"+1 matrix "B" with rows indexed by elements of GF(2)"n" and columns indexed by integers 0..."n", and entries
The sum of the rows of "B" is "M" times the inner distribution vector ("A"0...,"A""n").
A code "C" is regular if the rows of "B" corresponding to the codewords of "C" are all equal.