Skew polynomial representations of matrix algebras and applications to coding theory
Alessandro Neri, Paolo Santonastaso
TL;DR
The paper develops a broad skew-polynomial framework to represent direct sums of matrix algebras and to quantify sum-rank distance via an F-weight on skew polynomials. It proves an isometry between sum-rank metric codes and quotient skew-polynomial rings, enabling unified constructions of MSRD codes that generalize linearized Reed–Solomon, twisted variants, MRD codes, and Hamming-optimal codes. It then specializes to finite fields, deriving explicit length bounds and two new MDS families in the Hamming metric, while also extending MSRD constructions to infinite fields and to noncommutative division rings. The results yield infinitely many genuinely new MSRD codes, characterized by their nuclear parameters, and establish inequivalence with previously known families for broad parameter regimes.
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.