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 minimality of certain constructions.

翻译：本文的主要目的是进一步研究最近在和秩度量中引入的极小码的结构、参数和构造方法。这类对象构成了汉明度量中经典极小码（过去三十年来由于其密码学特性而受到广泛研究）与更近期的秩度量极小码之间的桥梁。我们证明了其参数的一些界值、存在性结果，并通过一种称为几何对偶的工具，成功构造了具有较少权重的极小码。本文还证明了著名的Ashikhmin-Barg条件的一个推广形式，并将其用于确保某些构造的极小性。