Generalized Hyperderivative Reed-Solomon Codes

TL;DR

This paper introduces Generalized Hyperderivative Reed-Solomon codes, expanding Reed-Solomon codes to include new properties like being MDS, duals, LDPC, and quasi-cyclic, with broad applicability.

Contribution

It generalizes NRT Reed-Solomon codes to GHRS codes, establishing their MDS property, duality, and identifying subfamilies with LDPC and quasi-cyclic structures.

Findings

GHRS codes are MDS

Duals of GHRS codes are also GHRS

Identified GHRS subfamilies as LDPC and quasi-cyclic

Abstract

This article introduces Generalized Hyperderivative Reed-Solomon codes (GHRS codes), which generalize NRT Reed-Solomon codes. Its main results are as follows: 1) every GHRS code is MDS, 2) the dual of a GHRS code is also an GHRS code, 3) determine subfamilies of GHRS codes whose members are low-density parity-check codes (LDPCs), and 4) determine a family of GHRS codes whose members are quasi-cyclic. We point out that there are GHRS codes having all of these properties.