Twisted Reed-Solomon Codes

TL;DR

This paper introduces twisted Reed-Solomon codes, a new class of evaluation codes that can be MDS and are often distinct from generalized RS codes, with practical decoding demonstrated through extensive simulations.

Contribution

It presents a novel construction of evaluation codes called twisted Reed-Solomon codes, expanding the family of MDS codes and analyzing their properties and decoding methods.

Findings

Several families of MDS codes are obtained.

Twisted RS codes are often not generalized RS codes.

The proposed decoder performs well in practice.

Abstract

In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed-Solomon codes. Whereas Reed-Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results.