The geometry of covering codes in the sum–rank metric

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-04-09 DOI:10.1007/s10623-025-01628-6

Matteo Bonini, Martino Borello, Eimear Byrne

引用次数: 0

Abstract

We introduce the concept of a sum–rank saturating system and outline its correspondence to covering properties of a sum–rank metric code. We consider the problem of determining the shortest length of a sum–rank-\(\rho \)-saturating system of a fixed dimension, which is equivalent to the covering problem in the sum–rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.

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和秩度量中覆盖码的几何形状

我们引入了和秩饱和系统的概念，并概述了它与和秩度量码的覆盖性质的对应关系。我们考虑确定固定维的和秩- \(\rho \) -饱和系统的最短长度问题，它等价于和秩度量中的覆盖问题。我们得到了这个量的上界和下界。我们还给出了由几何结构引起的饱和系统的构造。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.