MDS codes play a central role in practice due to their broad applications. To date, most known MDS codes are generalized Reed-Solomon (GRS) codes, leaving codes that are not equivalent to GRS codes comparatively less understood. Studying this non-GRS regime is therefore of intrinsic theoretical interest, and is also practically relevant since the strong algebraic structure of GRS codes can be undesirable in cryptographic settings. Among the known non-GRS codes, twisted generalized Reed-Solomon (TGRS) codes and Roth-Lempel codes are two representative families of non-GRS codes that have attracted significant attention. Though substantial work has been devoted to the construction and structural analysis of TGRS and Roth-Lempel codes, comparatively little attention has been paid to their decoding, and many problems remain open. In this paper, we propose list and unique decoding algorithms for TGRS codes and Roth-Lempel codes based on the Guruswami-Sudan algorithm. Under suitable parameter conditions, our algorithms achieve near-linear running time in the code length, improving upon the previously best-known quadratic-time complexity. Our TGRS decoder supports fixed-rate TGRS codes with up to O(n^2) twists, substantially extending prior work that only handled the single-twist case. For Roth-Lempel codes, we provide what appears to be the first efficient decoder. Moreover, our list decoders surpass the classical unique-decoding radius for a broad range of parameters. Finally, we incorporate algebraic manipulation detection (AMD) codes into the list-decoding framework, enabling recovery of the correct message from the output list with high probability.

翻译：MDS码因其广泛应用在实践中占据核心地位。迄今为止，已知的MDS码大多为广义Reed-Solomon（GRS）码，而不同于GRS码的码字相对理解较少。因此，研究这类非GRS码具有内在的理论意义，且在密码学场景中，GRS码强代数结构的不良特性也使其具有实际价值。在已知的非GRS码中，扭曲广义Reed-Solomon（TGRS）码和Roth-Lempel码是两类代表性非GRS码，已引起广泛关注。尽管大量工作致力于TGRS码和Roth-Lempel码的构造与结构分析，但其译码研究相对薄弱，且许多问题仍未解决。本文基于Guruswami-Sudan算法，提出针对TGRS码和Roth-Lempel码的列表译码与唯一译码算法。在合适的参数条件下，我们的算法实现了接近线性的码长时间复杂度，优于先前最优的二次复杂度。对于TGRS码，我们的译码器支持固定码率且含多达O(n²)个扭曲的码字，显著扩展了仅处理单扭曲情形的先前工作。针对Roth-Lempel码，我们提供了首个高效译码器。此外，我们的列表译码器在广泛参数范围内超越了经典唯一译码半径。最后，我们将代数操作检测（AMD）码融入列表译码框架，能够以高概率从输出列表中恢复正确消息。