Maximum Sum-Rank Distance Codes over Finite Chain Rings

TL;DR

This paper extends maximum sum-rank distance codes and linearized Reed-Solomon codes to finite chain rings, providing new theoretical results, decoding algorithms, and applications in space-time and network coding.

Contribution

It proves linearized Reed-Solomon codes are MSRD over finite chain rings and develops new decoding algorithms, extending prior finite field results.

Findings

Linearized Reed-Solomon codes are MSRD over finite chain rings.

A sum-rank Welch-Berlekamp decoder with cubic complexity is proposed.

A sum-rank syndrome decoder with quadratic complexity is introduced, first for these codes over finite fields.

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 minimum-degree annihilator skew polynomials and Lagrange interpolator skew polynomials. A general cubic-complexity sum-rank Welch-Berlekamp decoder and a quadratic-complexity sum-rank syndrome decoder (under some assumptions) are then provided over finite chain rings. The latter also constitutes the first known syndrome decoder for linearized Reed--Solomon codes over finite fields. Finally, applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding are discussed.