Maximum Sum-Rank Distance Codes over Finite Chain Rings

Abstract

In this work, maximum sum-rank distance (MSRD) codes and linearized Reed-Solomon codes are extended to finite chain rings. It is proven that linearized Reed-Solomon codes are MSRD over finite chain rings, extending the known result for finite fields. For the proof, several results on the roots of skew polynomials are extended to finite chain rings. These include the existence and uniqueness of minimumdegree annihilator skew polynomials and Lagrange interpolator skew polynomials. An efficient Welch-Berlekamp decoder with respect to the sum-rank metric is then provided for finite chain rings. Finally, applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding are discussed.