On q-Ary Shortened-1-Perfect-Like Codes

被引：1

作者：

Shi, Minjia ^{[1
]}

Wu, Rongsheng ^{[2
]}

Krotov, Denis S. ^{[3
]}

机构：

[1] Anhui Univ, Sch Math Sci, Key Lab Intelligent Comp & Signal Proc, Minist Educ, Hefei 230601, Anhui, Peoples R China

[2] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Anhui, Peoples R China

[3] Sobolev Inst Math, Novosibirsk 630090, Russia

来源：

IEEE TRANSACTIONS ON INFORMATION THEORY | 2022年 / 68卷 / 11期

基金：

中国国家自然科学基金; 中国博士后科学基金;

关键词：

Hamming graph; multifold packings; multiple coverings; perfect codes; PERFECT CODES; BINARY-CODES; PARAMETERS;

D O I：

10.1109/TIT.2022.3187004

中图分类号：

TP [自动化技术、计算机技术];

学科分类号：

0812 ;

摘要：

We study codes with parameters of q-ary shortened Hamming codes, i.e., (n = (q(m) - q) / (q - 1), q(n-m), 3)(q). Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal q-fold packing with minimum distance 2. Secondly, for every admissible length starting from n = 20, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.

引用

页码：7100 / 7106

页数：7

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