Decoding Reed-Solomon Skew-Differential Codes

TL;DR

This paper introduces a new class of MDS linear codes called Reed-Solomon skew-differential codes, with accessible linear algebra-based definitions and efficient decoding algorithms, and situates them within skew polynomial ring frameworks.

Contribution

The paper presents a novel class of Reed-Solomon skew-differential codes with simple linear algebra-based construction and decoding, expanding the theory of codes over skew polynomial rings.

Findings

Constructed a large class of MDS linear codes

Developed an efficient decoding algorithm based on linear algebra

Located these codes within the framework of skew polynomial rings

Abstract

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accesible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings.