Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

Abstract

Reed–Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed–Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed–Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed–Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed–Solomon codes and the sum-rank metric. The theories of Reed–Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before.