On the packing density of Lee spheres

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

Designs, Codes and Cryptography Pub Date : 2024-04-30 DOI:10.1007/s10623-024-01410-0

Ang Xiao, Yue Zhou

引用次数: 0

Abstract

Based on the packing density of cross-polytopes in \({\mathbb {R}}^n\), more than 50 years ago Golomb and Welch proved that the packing density of Lee spheres in \({\mathbb {Z}}^n\) must be strictly smaller than 1 provided that the radius r of the Lee sphere is large enough compared with n, which implies that there is no perfect Lee code for the corresponding parameters r and n. In this paper, we investigate the lattice packing density of Lee spheres with fixed radius r for infinitely many n. First we present a method to verify the nonexistence of the second densest lattice packing of Lee spheres of radius 2. Second, we consider the constructions of lattice packings with density \(\delta _n\rightarrow \frac{2^r}{(2r+1)r!}\) as \(n\rightarrow \infty \). When \(r=2\), the packing density can be improved to \(\delta _n\rightarrow \frac{2}{3}\) as \(n\rightarrow \infty \).

Abstract Image

查看原文本刊更多论文

关于李球的堆积密度

基于 \({\mathbb {R}}^n\) 中交叉多面体的堆积密度，50 多年前 Golomb 和 Welch 证明了只要李球的半径 r 与 n 相比足够大，那么 \({\mathbb {Z}}^n\) 中李球的堆积密度一定严格小于 1，这意味着对于相应的参数 r 和 n，不存在完美的李码。首先，我们提出了一种方法来验证半径为 2 的李球的第二最密晶格堆积的不存在性。其次，我们考虑密度为 \(\delta _n\rightarrow \frac{2^r}{(2r+1)r!}\) 的晶格堆积的构造。当 \(r=2\) 时，堆积密度可以改进为 \(\delta _n\rightarrow \frac{2}{3}\) 如 \(n\rightarrow \infty \)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

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.