On the weight distribution of the cosets of MDS codes

TL;DR

This paper refines the understanding of the weight distribution of cosets in MDS codes, providing new formulas and classifications for different coset weights, with applications to projective geometry and deep hole analysis.

Contribution

It introduces a structured formula for coset weight distributions, simplifies calculations for specific weights, and connects these distributions to geometric and covering properties.

Findings

All cosets of weight W=1 and W=d-1 have identical distributions.

Distributions for W=2 and W=d-2 are symmetrical.

MDS codes of covering radius d-1 relate to deep holes and optimal coverings.

Abstract

The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance $d$ using the known numbers of vectors of weights $\le d-2$ in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights $W$. (The weight $W$ of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered $W$ or regions of $W$, special relations more simple than the general ones are obtained. For the MDS code cosets of weight $W=1$ and weight $W=d-1$ we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight $W=1$ (as well as $W=d-1$) have the same weight distribution. The cosets of weight $W=2$ or $W=d-2$ may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane $\mathrm{PG}(2,q)$ are also considered. For MDS codes of covering radius $R=d-1$ we obtain the number of the weight $W=d-1$ cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius $R=d-1$ is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space $\mathrm{PG}(N,q)$.