On linear diameter perfect Lee codes with distance 6

IF 0.9 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2023-09-22 DOI:10.1016/j.jcta.2023.105816

Tao Zhang , Gennian Ge

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引用次数: 1

Abstract

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius $r \geq 2$ and dimension $n \geq 3$ . A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the $D P L (3, 6)$ code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no $D P L (n, d)$ codes for dimension $n \geq 3$ and distance $d > 4$ except for $(n, d) = (3, 6)$ . In this paper, we give a counterexample to this conjecture. Moreover, we prove that for $n \geq 3$ , there is a linear $D P L (n, 6)$ code if and only if $n = 3, 11$ .

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关于距离为6的线性直径完美Lee码

1968年，Golomb和Welch猜想不存在半径r≥2、维数n≥3的完美李码。直径完美码是完美码的自然推广。2011年，Etzion（2011）[5]提出了以下问题：除了DPL（3,6）码之外，是否存在距离大于4的直径完美Lee（简称DPL）码？后来，Horak和AlBdaiwi（2012）[12]推测，对于维数n≥3和距离d>；4，除了（n，d）=（3,6）。在本文中，我们给出了一个反例。此外，我们证明了对于n≥3，存在线性DPL（n，6）码当且仅当n=3,11。

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

Journal of Combinatorial Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.