Twisted Linearized Reed-Solomon Codes: A Skew Polynomial Framework

TL;DR

This paper introduces a new algebraic framework for sum-rank metric codes using skew polynomial rings, leading to the construction of twisted linearized Reed-Solomon codes that extend existing maximum distance codes in various metrics.

Contribution

It develops a skew polynomial framework for sum-rank codes and constructs twisted linearized Reed-Solomon codes, generalizing and extending known maximum distance codes in multiple metrics.

Findings

Constructed twisted linearized Reed-Solomon codes with maximum sum-rank distance.

Extended the framework to include an analogue of Trombetti-Zhou construction.

Discovered new additive MDS codes over quadratic fields in the Hamming metric case.

Abstract

We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.