weight enumerator

Let A be an alphabet and C a finite subset of A*. Then the complete weight enumerator of C, denoted by cweC, is the polynomialPlanetmathPlanetmath in |A| indeterminates Xa labeled by the letters of aA with integer coefficients defined by


where wta(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 weC, as a polynomial in only two indeterminates X and Y:

weC(X,Y):=cweC((Xa)aA)|X0=XXa=Y if a0,

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 cweC and weC are http://planetmath.org/node/6577homogeneous of degree n. Therefore, one can set Y=1 in weC in this case without losing information. The resulting polynomial can be uniquely rewritten in the form


the sequence A0,An defining the Hamming weight distribution. Analogously, one can define more general weight distributions by setting all but one indeterminate in cweC((Xa)aA) equal to one.


  • Let C be the ternary (that is A=𝔽3={0,1,2}) linear codeMathworldPlanetmath of block length 4 the vectors (1,1,1,1), (1,1,0,0) and (1,0,1,0). Then




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

  • The Hamming weight enumerator of the full binary code of length n, 𝔽2n, is simply given by we𝔽2n(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