The extended code for a class of generalized Roth-Lempel codes and their properties

TL;DR

This paper introduces extended generalized Roth-Lempel (EGRL) codes, analyzes their properties, and constructs NMDS EGRL codes with known weight distribution, expanding the understanding of code modifications and their optimality.

Contribution

It defines a new class of extended codes based on GRL codes, provides conditions for their MDS/AMDS properties, and constructs NMDS EGRL codes with explicit weight distribution.

Findings

Provided a parity-check matrix for a special class of EGRL codes.

Established necessary and sufficient conditions for EGRL codes to be MDS or AMDS.

Constructed NMDS EGRL codes and determined their weight distribution.

Abstract

As we all know, many interesting and important codes are obtained by modifying or combining existing codes. In this paper, we focus on generalized Roth-Lempel (in short, GRL) codes and define a class of extended codes, i.e., the extended generalized Roth-Lempel (in short, EGRL) code. And then for a special class of EGRL codes, we give a parity-check matrix and establish a necessary and sufficient condition for the EGRL code or its dual code to be MDS or AMDS, respectively. Finally, we construct a class of NMDS EGRL codes which is the generalization of the constructions given by Han et al. in 2023, and then completely determine its weight distribution.