New bounds for $b$-Symbol Distances of Matrix Product Codes

TL;DR

This paper extends bounds on the $b$-symbol distances of matrix product codes, providing new theoretical insights and constructions for codes with optimal or near-optimal $b$-symbol distances, including Reed-Muller and almost MDS codes.

Contribution

It generalizes existing bounds to $b$-symbol distances, determines minimum $b$-symbol distances for several code families, and constructs new $b$-symbol almost MDS codes.

Findings

Generalized bounds for $b$-symbol distances of matrix product codes.

Determined minimum $b$-symbol distances for Reed-Muller, $[u+v,u-v]$-construction, and specific linear codes.

Constructed new $b$-symbol almost MDS codes with arbitrary length.

Abstract

Matrix product codes are generalizations of some well-known constructions of codes, such as Reed-Muller codes, $[u+v,u-v]$-construction, etc. Recently, a bound for the symbol-pair distance of a matrix product code was given in \cite{LEL}, and new families of MDS symbol-pair codes were constructed by using this bound. In this paper, we generalize this bound to the $b$-symbol distance of a matrix product code and determine all minimum $b$-symbol distances of Reed-Muller codes. We also give a bound for the minimum $b$-symbol distance of codes obtained from the $[u+v,u-v]$-construction, and use this bound to construct some $[2n,2n-2]_q$-linear $b$-symbol almost MDS codes with arbitrary length. All the minimum $b$-symbol distances of $[n,n-1]_q$-linear codes and $[n,n-2]_q$-linear codes for $1\leq b\leq n$ are determined. Some examples are presented to illustrate these results.