weight enumerator

Let $A$ be an alphabet and $C$ a finite subset of $A^{*}$ . Then the complete weight enumerator of $C$ , denoted by $\operatorname{cwe}_{C}$ , is the polynomial in $|A|$ indeterminates $X_{a}$ labeled by the letters of $a\in A$ with integer coefficients defined by

\operatorname{cwe}_{C}((X_{a})_{a\in A}):=\sum\limits_{c\in C}\prod\limits_{a% \in A}X_{a}^{\operatorname{wt}_{a}(c)},

where $\operatorname{wt}_{a}(c)$ is the $a$ -weight of the string $c$ .

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

\operatorname{we}_{C}(X,Y):=\operatorname{cwe}_{C}((X_{a})_{a\in A})|_{\begin{% array}[]{l}\scriptstyle X_{0}=X\\ \scriptstyle X_{a}=Y\text{ if }a\neq 0\end{array}},

that is one distinguishes only between zero and the non-zero letters of the strings in $C$ .

If $C$ is a code of block length $n$ , then both $\operatorname{cwe}_{C}$ and $\operatorname{we}_{C}$ are http://planetmath.org/node/6577homogeneous of degree $n$ . Therefore, one can set $Y=1$ in $\operatorname{we}_{C}$ in this case without losing information. The resulting polynomial can be uniquely rewritten in the form

\operatorname{we}_{C}(X,1)=\sum\limits_{i=0}^{n}A_{i}X^{n-i},

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 $\operatorname{cwe}_{C}((X_{a})_{a\in A})$ equal to one.

Examples

•

Let $C$ be the ternary (that is $A=\mathbb{F}_{3}=\{0,1,2\}$ ) linear code of block length $4$ the vectors $(1,1,1,1)$ , $(1,1,0,0)$ and $(1,0,1,0)$ . Then

$\operatorname{cwe}_{C}(X_{0},X_{1},X_{2})=X_{0}^{4}+4X_{0}^{2}X_{1}^{2}+4X_{0}% ^{2}X_{1}X_{2}+4X_{0}^{2}X_{2}^{2}+4X_{0}X_{1}^{2}X_{2}+4X_{0}X_{1}X_{2}^{2}+X% _{1}^{4}+4X_{1}^{2}X_{2}^{2}+X_{2}^{4}$

and

$\operatorname{we}_{C}(X,Y)=X^{4}+12X^{2}Y^{2}+8XY^{3}+6Y^{4}$

and the Hamming weight distribution is $1,0,12,8,6$ .
•

The Hamming weight enumerator of the full binary code of length $n$ , $\mathbb{F}_{2}^{n}$ , is simply given by $\operatorname{we}_{\mathbb{F}_{2}^{n}}(X,Y)=(X+Y)^{n}$ , and the Hamming weight distribution is the $n$ -th of Pascal’s triangle.

Title	weight enumerator
Canonical name	WeightEnumerator
Date of creation	2013-03-22 15:13:23
Last modified on	2013-03-22 15:13:23
Owner	GrafZahl (9234)
Last modified by	GrafZahl (9234)
Numerical id	4
Author	GrafZahl (9234)
Entry type	Definition
Classification	msc 94A55
Classification	msc 94B05
Synonym	Hamming weight enumerator
Related topic	KleeneStar
Related topic	LinearCode
Defines	complete weight enumerator
Defines	weight distribution
Defines	Hamming weight distribution