MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing $[n,k]$ MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at most $k-1$ zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.

翻译：MDS码因其在实际中的广泛应用而受到极大关注。迄今为止，大多数已知的MDS码均等价于里德-所罗门码。非里德-所罗门型MDS码的构造已成为编码理论与有限几何领域中一个引人入胜且至关重要的问题。尽管文献中已提出若干非里德-所罗门型MDS码的构造方法，但这些MDS码的参数仍受严格限制。本文通过选取合适的评估多项式集与评估点集，使得所有非零多项式在评估集中的零点数不超过$k-1$，从而提出构建$[n,k]$ MDS码的通用框架。此外，通过计算这些码的舒尔平方可证明其非里德-所罗门特性。进一步地，通过转化为组合问题，本文给出了若干非里德-所罗门MDS码的显式构造。由此可获得长度灵活性显著提升的新型非里德-所罗门MDS码族，其中绝大多数码族未被现有研究成果所覆盖。