Maximum Sum-Rank Distance Codes over Finite Chain Rings
Umberto Martínez-Peñas, Sven Puchinger
TL;DR
The paper extends maximum sum-rank distance (MSRD) codes and linearized Reed–Solomon codes to finite chain rings, establishing that linearized Reed–Solomon codes are MSRD over these rings by extending skew-polynomial root theory. It develops foundational algebraic tools (annihilator and Lagrange interpolators, extended Moore matrices) for skew polynomials on finite chain rings and uses them to construct MSRD linearized Reed–Solomon codes with explicit MSRD parameter sets. Two decoding approaches are provided: a cubic-complexity Welch–Berlekamp decoder valid in general, and a quadratic-complexity syndrome decoder that applies under a coprimality condition, with implications including the first known syndrome decoder for LRS codes over finite fields. The work also discusses practical applications in Space-Time Coding with multiple fading blocks and physical-layer multishot Network Coding, illustrating how ring-based alphabets and morphisms to complex constellations enable robust, efficient coding in these communication scenarios.
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.