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.

翻译：本文将对最大和秩距（MSRD）码及线性化Reed-Solomon码推广到有限链环上。证明线性化Reed-Solomon码在有限链环上仍保持MSRD性质，从而扩展了已知的有限域结论。为完成证明，将有限域上关于斜多项式根的若干结果推广到有限链环，包括最小次数零化斜多项式的存在唯一性及Lagrange插值斜多项式。随后给出有限链环上的通用三次复杂度和秩Welch-Berlekamp译码器，以及（在某些假设下）二次复杂度和秩伴随式译码器。后者亦构成有限域上线性化Reed-Solomon码的首个伴随式译码器。最后讨论该方法在多衰落块空时编码及物理层多跳网络编码中的应用。