Weight Distributions

In the case of a linear code, weight and distance distributions are equivalent (in particular minimum weight and minimum distance are equivalent).

Contents

Hamming Weight

For an element x∈R for any finite ring R, the Hamming weight wH(x) is defined by: wH(x) = 0 iff x = 0, qquad wH(x) = 1 iff x ≠0 The Hamming weight wH(v) of a vector v∈(Rn) is defined to be the sum (in Z) of the Hamming weights of its components.

The Hamming weight is often referred to as simply the weight.

MinimumWeight(C) : Code -> RngIntElt
MinimumDistance(C) : Code -> RngIntElt
Determine the minimum (Hamming) weight of the words belonging to the code C, which is also the minimum distance between any two codewords.
WeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
Determine the (Hamming) weight distribution for the code C. The distribution is returned in the form of a sequence of tuples, where the i-th tuple contains the i-th weight, wi say, and the number of codewords having weight wi.
DualWeightDistribution(C) : Code -> [ <RngIntElt, RngIntElt> ]
Determine the (Hamming) weight distribution of the dual code of C. The distribution is returned in the form of a sequence of tuples, where the i-th tuple contains the i-th weight, wi say, and the number of codewords having weight wi.

Example CodeRng_weight-dist-cyc (H164E14)

We calculate the weight distribution of a cyclic code over the Galois ring of size 81. 
> R<w> := GR(9,2);
> P<x> := PolynomialRing(R);
> L := CyclotomicFactors(R, 4);
> g := L[3] * L[4];
> g;
x^2 + (8*w + 7)*x + w + 1
> C := CyclicCode(4, g);
> C;
(4, 6561, 3) Cyclic Code over GaloisRing(3, 2, 2)
Generator matrix:
[      1       0   w + 1 8*w + 7]
[      0       1       w 8*w + 8]
> WeightDistribution(C);
[ <0, 1>, <3, 320>, <4, 6240> ]
V2.28, 13 July 2023