A general family of MSRD codes and PMDS codes with smaller field sizes from extended Moore matrices

TL;DR

This paper introduces six new explicit families of MSRD codes with smaller field sizes, and two new families of PMDS codes, achieved through innovative construction methods involving extended Moore matrices and classical codes.

Contribution

The paper presents novel explicit constructions of MSRD and PMDS codes with reduced field sizes, expanding the known parameter regimes and improving practical applicability.

Findings

Six new MSRD code families with minimal field sizes for certain parameters.

Two new PMDS code families with smaller fields than existing codes.

MSRD codes based on Hamming codes meet a recent theoretical bound.

Abstract

We construct six new explicit families of linear maximum sum-rank distance (MSRD) codes, each of which has the smallest field sizes among all known MSRD codes for some parameter regime. Using them and a previous result of the author, we provide two new explicit families of linear partial MDS (PMDS) codes with smaller field sizes than previous PMDS codes for some parameter regimes. Our approach is to characterize evaluation points that turn extended Moore matrices into the parity-check matrix of a linear MSRD code. We then produce such sequences from codes with good Hamming-metric parameters. The six new families of linear MSRD codes with smaller field sizes are obtained using MDS codes, Hamming codes, BCH codes and three Algebraic-Geometry codes. The MSRD codes based on Hamming codes, of minimum sum-rank distance $ 3 $, meet a recent bound by Byrne et al.