Generalized Reed-Solomon codes form the most prominent class of maximum distance separable (MDS) codes, codes that are optimal in the sense that their minimum distance cannot be improved for a given length and code size. The study of codes that are MDS yet not generalized Reed-Solomon codes, called non-generalized Reed-Solomon MDS codes, started with the work by Roth and Lemple (1989), where the first examples where exhibited. It then gained traction thanks to the work by Beelen (2017), who introduced twisted Reed-Solomon codes, and showed that families of such codes are non-generalized Reed-Solomon MDS codes. Finding non-generalized Reed-Solomon MDS codes is naturally motivated by the classification of MDS codes. In this paper, we provide a generic construction of MDS codes, yielding infinitely many examples. We then explicit families of non-generalized Reed-Solomon MDS codes. Finally we position some of the proposed codes with respect to generalized twisted Reed-Solomon codes, and provide new view points on this family of codes.

翻译：广义Reed-Solomon码是最大距离可分码中最重要的一类，这类码在给定长度和码本大小的条件下具有最优的最小距离特性。对于具有MDS特性但非广义Reed-Solomon码的编码（称为非广义Reed-Solomon MDS码）的研究始于Roth和Lemple（1989）的工作，其中首次展示了相关实例。随后Beelen（2017）通过引入扭曲Reed-Solomon码并证明其若干子类属于非广义Reed-Solomon MDS码，推动了该领域的发展。寻找非广义Reed-Solomon MDS码的动机源于对MDS码分类的理论需求。本文提出了一种通用的MDS码构造方法，可生成无限多实例。我们进而显式构造了若干非广义Reed-Solomon MDS码族。最后，我们将部分所提编码与广义扭曲Reed-Solomon码进行对比定位，并为该类编码提供了新的研究视角。