Constructions of non-Generalized Reed-Solomon MDS codes
Shengwei Liu, Hongwei Liu, Frederique Oggier
TL;DR
This work addresses the problem of classifying MDS codes that are not generalized Reed-Solomon codes. It introduces a Vandermonde-based generic construction that yields infinitely many MDS codes and explicitly constructs non-GRS MDS families using controlled perturbations and extension-field twists. It then analyzes the relationship with generalized twisted Reed-Solomon (GTRS) codes, showing that some non-GRS examples can still be GTRS and providing criteria and examples, while noting that determining non-GTRS status remains open. Overall, the paper expands the catalog of MDS codes and advances the understanding of how non-GRS MDS codes relate to GTRS codes, with implications for code classification and design.
Abstract
Generalized Reed-Solomon codes form the most prominent class of maximum distance separable (MDS) codes, codes that are optimal in the sense that their minimum distance cannot be improved for a given length and code size. The study of codes that are MDS yet not generalized Reed-Solomon codes, called non-generalized Reed-Solomon MDS codes, started with the work by Roth and Lemple (1989), where the first examples where exhibited. It then gained traction thanks to the work by Beelen (2017), who introduced twisted Reed-Solomon codes, and showed that families of such codes are non-generalized Reed-Solomon MDS codes. Finding non-generalized Reed-Solomon MDS codes is naturally motivated by the classification of MDS codes. In this paper, we provide a generic construction of MDS codes, yielding infinitely many examples. We then explicit families of non-generalized Reed-Solomon MDS codes. Finally we position some of the proposed codes with respect to generalized twisted Reed-Solomon codes, and provide new view points on this family of codes.