Cyclic maximum distance separable (MDS for short) codes are a special subclass of linear codes and have received a lot of attention, as these codes have very important applications in many areas including quantum codes, designs and finite geometry. However, the existing construction methods for cyclic MDS codes are mainly focused on strict restrictions on certain parameters or are relatively complex in their construction approaches. In this paper, we investigate this approach further via norm reduction in cyclotomic fields and present a construction method of cyclic MDS codes over finite fields. We transform the problem of verifying the MDS property over a finite field into a problem of determining non-zero minors in characteristic zero. Compared with existing construction methods, our method is relatively simple. In particular, the results of this paper show that the parameters of non-RS cyclic MDS codes are flexible and completely cover the results in [Non-Reed-Solomon Type Cyclic MDS codes, IEEE Trans. Inf. Theory, 71(5): 3489--3496, 2025].

翻译：循环最大距离可分（简称MDS）码是线性码的一个特殊子类，由于其在量子码、设计和有限几何等多个领域中具有非常重要的应用而受到了广泛关注。然而，现有的循环MDS码构造方法主要局限于对特定参数的严格限制，或者其构造途径相对复杂。本文通过分圆域中的范数降阶方法进一步研究这一途径，并提出了一种在有限域上构造循环MDS码的方法。我们将有限域上验证MDS性质的问题转化为特征为零时确定非零子式的问题。与现有的构造方法相比，我们的方法相对简单。特别地，本文的结果表明非RS类型循环MDS码的参数是灵活的，并且完全覆盖了[Non-Reed-Solomon Type Cyclic MDS codes, IEEE Trans. Inf. Theory, 71(5): 3489--3496, 2025]中的结果。