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

TL;DR

This paper introduces linearized Reed-Solomon codes over division rings, demonstrating they achieve maximum sum-rank distance, extending classical Reed-Solomon and Gabidulin codes to new algebraic settings with applications in network coding.

Contribution

It presents a new class of codes called linearized Reed-Solomon codes, proving their maximum sum-rank distance over any division ring, generalizing existing codes and metrics.

Findings

Linearized Reed-Solomon codes achieve maximum sum-rank distance.

The theory extends classical Reed-Solomon and Gabidulin codes.

New maximum rank distance codes over infinite fields with non-zero derivations.

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.