MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and establish some sufficient and necessary conditions for them being MDS. Notably, these codes differ from Reed-Solomon codes up to monomial equivalence. Additionally, we also explore the cases in which these codes are almost MDS or near MDS. Applying our main results, we determine the covering radii and deep holes of the dual codes associated with specific Roth-Lempel codes and discover an infinite family of (almost) optimally extendable codes with dimension three.

翻译：MDS码因其代数性质与最优纠错能力，在通信系统、数据存储及量子编码等领域具有广泛实际应用。本文聚焦一类线性码，建立其成为MDS码的若干充要条件。值得关注的是，这些码在单项等价意义下不同于里德-所罗门码。此外，我们亦探讨了这些码成为几乎MDS或近MDS码的情形。应用主要结论，我们确定了特定Roth-Lempel码的对偶码的覆盖半径与深洞，并发现了一个包含三维(几乎)最优可扩展码的无限族。