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：本文研究了极大和秩距离（MSRD）码与线性化Reed-Solomon码在有限链环上的推广。证明了线性化Reed-Solomon码在有限链环上满足MSRD性质，推广了有限域上的已知结果。在证明过程中，将关于斜多项式根的若干结果推广至有限链环，包括最小次数零化斜多项式与拉格朗日插值斜多项式的存在唯一性。随后，在有限链环上分别提出了具有三次复杂度的和秩Welch-Berlekamp译码器以及（在特定假设下）二次复杂度的和秩伴随式译码器，后者同时构成了有限域上线性化Reed-Solomon码的首个已知伴随式译码器。最后，讨论了该方法在多衰落块空时编码与物理层多跳网络编码中的应用。