On q-ary shortened-1-perfect-like codes

On q-ary shortened-1-perfect-like codes Full article

Journal

IEEE Transactions on Information Theory
ISSN: 0018-9448 , E-ISSN: 1557-9654

Output data

Year: 2022, Volume: 68, DOI: 10.1109/TIT.2022.3187004

Tags

Binary codes; Codes; Hamming distance; Hamming graph; Mathematics; multifold packings; multiple coverings; perfect codes; Signal processing; Symbols; Upper bound

Authors

Shi M. ¹ , Wu R. ² , Krotov D.S. ³

Affiliations

1	School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, Anhui University, Hefei, Anhui, China
2	School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
3	Sobolev Institute of Mathematics

Funding (1)

Sobolev Institute of Mathematics

FWNF-2022-0017

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.

Shi M. , Wu R. , Krotov D.S.
On q-ary shortened-1-perfect-like codes
IEEE Transactions on Information Theory. 2022. V.68. DOI: 10.1109/TIT.2022.3187004 WOS Scopus РИНЦ OpenAlex

Submitted:	Jul 4, 2021
Accepted:	Jun 22, 2022
Published online:	Jul 12, 2022

Web of science:	WOS:000871032100012
Scopus:	2-s2.0-85134262861
Elibrary:	57965139
OpenAlex:	W3205622575

DB	Citing
Scopus	3
Web of science	1
OpenAlex	2
Elibrary	1