On $\ell$-MDS codes and a conjecture on infinite families of $1$-MDS codes

TL;DR

This paper solves a conjecture for 1-MDS codes, constructs infinite families with specific combinatorial properties, and explores general properties and bounds of $oldsymbol{ ext{ extit{ extbf{ extcolor{blue}{ ext{ell}}}-MDS}}}$ codes.

Contribution

It completely resolves a recent conjecture for 1-MDS codes, constructs new $ ext{ extit{ extbf{ extcolor{blue}{ ext{ell}}}}}$-MDS codes, and generalizes key properties and bounds of these codes.

Findings

Solved the conjecture for 1-MDS codes.

Constructed infinite families of 1-MDS codes with 2-designs.

Established bounds and characterizations for $ ext{ extit{ extbf{ extcolor{blue}{ ext{ell}}}}}$-MDS codes.

Abstract

The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a} generalization of maximum distance separable (MDS) codes {that} has attracted a lot of attention due to its applications in several areas such as secret sharing schemes, index coding problems, informed source coding problems, and combinatorial $t$-designs. In this paper, for $\ell=1$, we completely solve a conjecture recently proposed by Heng $et~al.$ (Discrete Mathematics, 346(10): 113538, 2023) and obtain infinite families of $1$-MDS codes with general dimensions holding $2$-designs. These later codes are also been proven to be optimal locally recoverable codes. For general {positive integers} $\ell$ and $\ell'$, we construct new $\ell$-MDS codes from known $\ell'$-MDS codes via some classical propagation rules involving the extended, expurgated, and $(u,u+v)$ constructions. Finally, we study some general results including characterization, weight distributions, and bounds on maximum lengths of $\ell$-MDS codes, which generalize, simplify, or improve some known results in the literature.