weight enumerator

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

$${\mathrm{cwe}}_C({(X_a)}_{a\in A}):=\sum _{c\in C}\prod _{a\in A}{X}_a^{{\mathrm{wt}}_a(c)},$$

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

If $A$ is an abelian group, one defines the Hamming weightenumerator of $C$, denoted by ${\mathrm{we}}_C$, as a polynomial in only twoindeterminates $X$ and $Y$:

$${\mathrm{we}}_C(X,Y):={{\mathrm{cwe}}_C({(X_a)}_{a\in A})|}_{\begin{array}{c}{X}_0=X\hfill \\ X_a=Y\text{ if }a\ne 0\hfill \end{array}},$$

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

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

$${\mathrm{we}}_C(X,1)=\sum _{i=0}^n{A}_i{X}^{n-i},$$

the sequence $A_0,\mathrm{\dots }{A}_n$ defining the Hamming weightdistribution. Analogously, one can define more general weightdistributions by setting all but one indeterminate in${\mathrm{cwe}}_C({(X_a)}_{a\in A})$ equal to one.

Examples

• Let $C$ be the ternary (that is $A={𝔽}_3=\{0,1,2\}$) linearcode of block length $4$ the vectors $(1,1,1,1)$, $(1,1,0,0)$and $(1,0,1,0)$. Then

  $${\mathrm{cwe}}_C(X_0,X_1,X_2)=X_0^4+4X_0^2{X}_1^2+4X_0^2{X}_1{X}_2+4X_0^2{X}_2^2+4X_0{X}_1^2{X}_2+4X_0{X}_1{X}_2^2+X_1^4+4X_1^2{X}_2^2+X_2^4$$

  and

  $${\mathrm{we}}_C(X,Y)=X^4+12X^2{Y}^2+8X{Y}^3+6Y^4$$

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

• The Hamming weight enumerator of the full binary code of length$n$, ${𝔽}_2^n$, is simply given by${\mathrm{we}}_{{𝔽}_2^n}(X,Y)={(X+Y)}^n$, and the Hamming weightdistribution 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