Skew polynomial representations of matrix algebras and applications to coding theory

Abstract We extend the existing skew polynomial representations of matrix algebras that 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 by‐product, in the case of finite fields, we obtain new families of maximum distance separable codes that are linear over a subfield and whose length is close to the field size.