On weight spectrum of linear codes

TL;DR

This paper investigates the conditions under which sequences of linear codes have a uniform weight spectrum, providing new examples and generalizations related to weight distribution, Fourier transform eigenfunctions, and MacWilliams identities.

Contribution

It establishes that a linear code sequence has a uniform weight spectrum if the number of weight-1 vectors grows infinitely, and presents a novel example with specific properties.

Findings

Sequences with infinite weight-1 vectors have uniform weight spectrum

Constructed example with half-length dimension lacks uniform spectrum

Generalized MacWilliams-type identity for weight distributions

Abstract

We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors from codes with weight $1$ grows to infinity. We find an example of a sequence of linear codes such that the dimension of the code is the half of the codelength but it has not a uniform weight spectrum. This example generates eigenfunctions of the Fourier transform with minimal support and partial covering sets. Moreover, we generalize some MacWilliams-type identity. Keywords: weight distribution of code, dual code, MacWilliams identity, Fourier transform, partial covering array