Let $A$ be an alphabet and $C$ a finite subset of $A^*$ . Then the complete weight enumerator of $C$ , denoted by $\cwe_C$ , is the polynomial in $|A|$ indeterminates $X_a$ labeled by the letters of $a\in A$ with integer coefficients defined by \begin{equation*} \cwe_C((X_a)_{a\in A}):=\Sum_{c\in C}\Prod_{a\in A}X_a^{\wt_a(c)}, \end{equation*}where $\wt_a(c)$ is the $a$ -weight of the string $c$ .

If $A$ is an abelian group, one defines the Hamming weight enumerator of $C$ , denoted by $\we_C$ , as a polynomial in only two indeterminates $X$ and $Y$ :

If $C$ is a code of block length $n$ , then both $\cwe_C$ and $\we_C$ are homogeneous of degree $n$ . Therefore, one can set $Y=1$ in $\we_C$ in this case without losing information. The resulting polynomial can be uniquely rewritten in the form \begin{equation*} \we_C(X,1)=\Sum_{i=0}^nA_iX^{n-i}, \end{equation*}the sequence $A_0,\ldots A_n$ defining the Hamming weight distribution. Analogously, one can define more general weight distributions by setting all but one indeterminate in $\cwe_C((X_a)_{a\in A})$ equal to one.

Examples

• Let $C$ be the ternary (that is $A=\mbb{F}_3=\{0,1,2\}$ ) linear code of block length $4$ spanned by the vectors $(1,1,1,1)$ , $(1,1,0,0)$ and $(1,0,1,0)$ . Then \begin{equation*} \cwe_C(X_0,X_1,X_2)=X_0^4+4X_0^2X_1^2+4X_0^2X_1X_2+4X_0^2X_2^2+4X_0X_1^2X_2+4X_0X_1X_2^2+X_1^4+4X_1^2X_2^2+X_2^4 \end{equation*}and \begin{equation*} \we_C(X,Y)=X^4+12X^2Y^2+8XY^3+6Y^4 \end{equation*}and the Hamming weight distribution is $1,0,12,8,6$ .
• The Hamming weight enumerator of the full binary code of length $n$ , $\mbb{F}_2^n$ , is simply given by $\we_{\mbb{F}_2^n}(X,Y)=(X+Y)^n$ , and the Hamming weight distribution is the $n$ -th row of Pascal's triangle.