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Reed-Solomon codes and Gabidulin codes have maximum Hamming distance andmaximum rank distance, respectively. A general construction using skewpolynomials, called skew Reed-Solomon codes, has already been introduced in theliterature. In this work, we introduce a linearized version of such codes,called linearized Reed-Solomon codes. We prove that they have maximum sum-rankdistance. Such distance is of interest in multishot network coding or insingleshot multi-network coding. To prove our result, we introduce new metricsdefined by skew polynomials, which we call skew metrics, we prove that skewReed-Solomon codes have maximum skew distance, and then we translate thisscenario to linearized Reed-Solomon codes and the sum-rank metric. The theoriesof 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 developour theory over any division ring (commutative or non-commutative field). Wealso consider non-zero derivations, which give new maximum rank distance codesover infinite fields not considered before.

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