Constructions of near MDS codes which are optimal locally recoverable codes

TL;DR

This paper introduces new constructions of near MDS (NMDS) codes with optimal local recoverability, explores their properties, and derives families of codes that are both distance- and dimension-optimal for data storage applications.

Contribution

The paper presents novel constructions of NMDS codes, analyzes their weight enumerators, and establishes their locality, leading to new families of optimal locally repairable codes.

Findings

Constructed several NMDS codes with diverse weight enumerators.

Determined the locality of NMDS codes.

Developed families of distance- and dimension-optimal locally repairable codes.

Abstract

A linear code with parameters $[n,k,n-k]$ is said to be almost maximum distance separable (AMDS for short). An AMDS code whose dual is also AMDS is referred to as an near maximum distance separable (NMDS for short) code. NMDS codes have nice applications in finite geometry, combinatorics, cryptography and data storage. In this paper, we first present several constructions of NMDS codes and determine their weight enumerators. In particular, some constructions produce NMDS codes with the same parameters but different weight enumerators. Then we determine the locality of the NMDS codes and obtain many families of distance-optimal and dimension-optimal locally repairable codes.