ECC dictionary W

weight enumerator (Complete):

Index the finite field GF(q) in some fixed way: GF(q) = { ω₀=0, ω₁, ... , ω_q-1 }. The composition of v = (v₁,..,v_n) in GF(q)ⁿ is defined by comp(v) = (s₀, ... ,s_q-1), where s_j=s_j(v) denotes the number of components of v equal to ω₀. Clearly, Σ_i=0^q-1 s_i(v)=n, for each v in GF(q)ⁿ. For s = (s₀, ... ,s_q-1) in ZZ^q, let T_C(s) denote the number of codewords c in C with comp(c)=s.

The complete weight enumerator polynomial A_C is defined by

W_C(z_{0, ...,} z_q-1) = Σ _i=0 W_C(z₀,\dots ,z_{q-1})
             = Σ _{c in C}   z₀^s₀(c) ... z_q-1^s_q-1(c)
             = Σ_{s in ZZ^q}   T_C(s)z₀^s₀ ... z_q-1^s_q-1.

Here ZZ denotes the ring of integers. 

Fact: If x =  z₀ and y = z₁  = ... = z_q-1  then W_C(z_{0, ...,} z_q-1) = A_C(x,y).

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weight enumerator (Hamming):

The (Hamming) weight enumerator polynomial A_C is defined by

A_C(x,y) = Σ _i=0 ⁿ A_i x^n-iyⁱ = xⁿ + A_d x^n-dy^d+ ... +A_n yⁿ,

where

A_i = |{c in C | wt(c)=i}|

denotes the number of codewords of weight i. Of course, A_C(1,1) = |C|.

See also weight enumerator at Wikipedia.