The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank-metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated Ashikhmin–Barg condition is proved and used to ensure the minimality of certain constructions.

中文翻译：

---

几何对偶和和秩最小码

本文的主要目的是进一步研究最近引入的和秩度量中的最小码的结构、参数和构造。这些对象在汉明度量中的经典最小代码和最近的秩度量最小代码之间架起了一座桥梁，汉明度量是过去三十年来深入研究的主题，部分原因是它们的密码特性。我们证明了它们的参数、存在结果的一些界限，并且通过我们称为几何对偶的工具，我们设法构建具有很少权重的最小代码。著名的 Ashikhmin-Barg 条件的推广被证明并用于确保某些结构的最小化。