Skew polynomial representations of matrix algebras and applications to coding theory

TL;DR

This paper extends skew polynomial representations of matrix algebras to construct new maximum sum-rank distance codes and MDS codes, broadening the scope of optimal coding strategies over various fields and division rings.

Contribution

It introduces a generalized skew polynomial representation for matrix algebras and uses it to develop new families of MSRD and MDS codes, unifying and extending existing coding constructions.

Findings

Constructed new MSRD codes over finite and infinite fields.

Derived new MDS codes linear over subfields with lengths close to field size.

Unified framework generalizing known rank and Hamming metric codes.

Abstract

We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight function on their associated skew polynomials, defined through degrees and greatest common right divisors with the polynomial that defines the representation. We exploit this representation to construct new families of maximum sum-rank distance (MSRD) codes over finite and infinite fields, and over division rings. These constructions generalize many of the known existing constructions of MSRD codes as well as of optimal codes in the rank and in the Hamming metric. As a byproduct, in the case of finite fields we obtain new families of MDS codes which are linear over a subfield and whose length is close to the field size.