# 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

## Contents

## Basic properties[edit]

## MacWilliams identity[edit]

Denote the dual code of by

(where denotes the vector dot product and which is taken over ).

The **MacWilliams identity** states that

The identity is named after Jessie MacWilliams.

## Distance enumerator[edit]

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.

## Source[edit]

## See Also on BitcoinWiki[edit]