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Enumerator polynomial

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

Contents

Let C ⊂ F 2 n be a binary linear code length n . The weight distribution is the sequence of numbers

A t = # { c ∈ 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 ) = ∑ w = 0 n A w x w y n − w .

Basic properties

  1. W ( C ; 0 , 1 ) = A 0 = 1
  2. W ( C ; 1 , 1 ) = ∑ w = 0 n A w = | C |
  3. W ( C ; 1 , 0 ) = A n = 1  if  ( 1 , … , 1 ) ∈ C    and  0  otherwise
  4. W ( C ; 1 , − 1 ) = ∑ w = 0 n A w ( − 1 ) n − w = A n + ( − 1 ) 1 A n − 1 + … + ( − 1 ) n − 1 A 1 + ( − 1 ) n A 0

MacWilliams identity

Denote the dual code of C ⊂ F 2 n by

C ⊥ = { x ∈ F 2 n ∣ ⟨ x , c ⟩ = 0   ∀ c ∈ C }

(where < , > denotes the vector dot product and which is taken over F 2 ).

The MacWilliams identity states that

W ( C ⊥ ; x , y ) = 1 ∣ C ∣ W ( C ; y − x , y + x ) .

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

A i = 1 M # { ( c 1 , c 2 ) ∈ C × C ∣ d ( c 1 , c 2 ) = i }

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

A ( C ; x , y ) = ∑ 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 ∈ C ∣ d ( c , x ) = i } .

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

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

References

Enumerator polynomial Wikipedia
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