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:本文件的主要目的是进一步研究最近采用的最低标准标准的结构、参数和结构,这些物体构成了哈明标准传统最低标准之间的桥梁,而哈明标准是过去三十年中密集研究的主题,部分是由于其加密特性,也是最近几级最低标准。 我们证明了这些标准参数、存在结果的一些界限,并且通过一个我们用几何双重名称命名的工具,我们设法建立了最低标准,但重量却很少。 已经证明并使用了庆贺的Ashikhmin-Barg条件的概括性,以确保某些建筑的最小性。</s>