An enumeration of 1-perfect ternary codes

被引：3

作者：

Shi, Minjia ^{[1
,2
]}

Krotov, Denis S. ^{[3
]}

机构：

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

[2] Xidian Univ, State Key Lab Integrated Serv Networks, Xian 710071, Peoples R China

[3] Sobolev Inst Math, Novosibirsk 630090, Russia

来源：

DISCRETE MATHEMATICS | 2023年 / 346卷 / 07期

基金：

中国国家自然科学基金;

关键词：

Perfect codes; Ternary codes; Concatenation; Switching; FULL-RANK TILINGS; ERROR-CORRECTING CODES; ARY QUASI-GROUPS; PERFECT CODES; CONSTRUCTION;

D O I：

10.1016/j.disc.2023.113437

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We study codes with parameters of the ternary Hamming (n = (3m - 1)/2, 3n-m, 3) code, i.e., ternary 1-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary 1-perfect codes of rank n -m + 1, count their number, and prove that all such codes can be obtained from each other by a sequence of two -coordinate switchings. We enumerate ternary 1-perfect codes of length 13 obtained by concatenation from codes of lengths 9 and 4; we find that there are 93241327 equivalence classes of such codes. (c) 2023 Elsevier B.V. All rights reserved.

引用

页数：16

共 48 条

[1] ASSMUS EF, 1992, DESIGNS THEIR CODES
[2] On the Ranks and Kernels Problem for Perfect Codes
S. V. Avgustinovich
F. I. Solov'eva
O. Heden
[J]. Problems of Information Transmission, 2003, 39 (4) : 341 - 345
[3] The classification of some perfect codes
Avgustinovich, SV
Heden, O
Solov'eva, FI
[J]. DESIGNS CODES AND CRYPTOGRAPHY, 2004, 31 (03) : 313 - 318
[4] Tilings of binary spaces
Cohen, G
Litsyn, S
Vardy, A
Zemor, G
[J]. SIAM JOURNAL ON DISCRETE MATHEMATICS, 1996, 9 (03) : 393 - 412
[5] Cohen G., 1997, North-Holland Mathematical Library, V54
[6] Colbourn C. J., 2006, HDB COMBINATORIAL DE, P102
[7] Full rank tilings of finite abelian groups
Dinitz, M
[J]. SIAM JOURNAL ON DISCRETE MATHEMATICS, 2006, 20 (01) : 160 - 170
[8] On perfect codes and tilings: Problems and solutions
Etzion, T
Vardy, A
[J]. SIAM JOURNAL ON DISCRETE MATHEMATICS, 1998, 11 (02) : 205 - 223
[9] NEW CONSTRUCTION OF GROUP AND NON-GROUP PERFECT CODES
HEDEN, O
[J]. INFORMATION AND CONTROL, 1977, 34 (04): : 314 - 323
[10] ON THE STRUCTURE OF NON-FULL-RANK PERFECT q-ARY CODES
Heden, Olof
Krotov, Denis S.
[J]. ADVANCES IN MATHEMATICS OF COMMUNICATIONS, 2011, 5 (02) : 149 - 156

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