Enumerator polynomial

Template:Short description In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight.

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

A_{t} = # {c \in C ∣ 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

W (C; x, y) = \sum_{w = 0}^{n} A_{w} x^{w} y^{n - w} .

Basic properties

$W (C; 0, 1) = A_{0} = 1$
$W (C; 1, 1) = \sum_{w = 0}^{n} A_{w} = | C |$
$W (C; 1, 0) = A_{n} = 1 if (1, \dots, 1) \in C and 0 otherwise$
$W (C; 1, - 1) = \sum_{w = 0}^{n} A_{w} (- 1)^{n - w} = A_{n} + (- 1)^{1} A_{n - 1} + \dots + (- 1)^{n - 1} A_{1} + (- 1)^{n} A_{0}$

Denote the dual code of $C \subset 𝔽_{2}^{n}$ by

C^{⊥} = {x \in 𝔽_{2}^{n} ∣ ⟨ x, c ⟩ = 0 \forall c \in C}

(where $⟨, ⟩$ denotes the vector dot product and which is taken over $𝔽_{2}$ ).

The MacWilliams identity states that

W (C^{⊥}; x, y) = \frac{1}{∣ C ∣} W (C; y - x, y + x) .

The identity is named after Jessie MacWilliams.

The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers

A_{i} = \frac{1}{M} # {(c_{1}, c_{2}) \in C \times C ∣ d (c_{1}, c_{2}) = i}

where i ranges from 0 to n. The distance enumerator polynomial is

A (C; x, y) = \sum_{i = 0}^{n} A_{i} x^{i} y^{n - i}

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 GF(2)ⁿ and columns indexed by integers 0...n, and entries

B_{x, i} = # {c \in C ∣ d (c, x) = i} .

The sum of the rows of B is M times the inner distribution vector (A₀,...,A_n).

A code C is regular if the rows of B corresponding to the codewords of C are all equal.