ℤ4上LCD循环码及其二元像

doi:10.12263/DZXB.20200588

电子学报 ›› 2021, Vol. 49 ›› Issue (11): 2284-2288.DOI: 10.12263/DZXB.20200588

$ℤ 4$ 上LCD循环码及其二元像

开晓山, 廖文敬

合肥工业大学数学学院，安徽合肥 230601

收稿日期:2020-06-17 修回日期:2021-05-23 出版日期:2021-11-25
- 作者简介:
- 开晓山（通信作者）　男，1975年生，安徽青阳人. 合肥工业大学数学学院教授，博士生导师，研究方向为编码理论与信息安全.E-mail:kxs6@sina.com
  廖文敬　女，1996年生，安徽宿州人. 合肥工业大学硕士研究生，主要研究方向为代数编码. E-mail:1846755441@qq.com
- 基金资助:
- 国家自然科学基金 (61972126); 中央高校基本科研业务费专项资金 (PA2019GDZC0097)

LCD Cyclic Codes over ℤ₄ and Binary Images

KAI Xiao-shan, LIAO Wen-jing

School of Mathematics, Hefei University of Technology, Hefei, Anhui 230601, China

Received:2020-06-17 Revised:2021-05-23 Online:2021-11-25 Published:2021-11-25
- Supported by:
- National Natural Science Foundation of China (61972126); Fundamental Research Funds for the Central Universities (PA2019GDZC0097)

摘要/Abstract

摘要：

循环码和线性互补对偶（LCD）码是两类重要的线性码，在数据存储、通信系统和密码等领域有着广泛的应用. 本文研究了 $? 4$ 上奇长度的LCD循环码，给出了 $? 4$ 上奇长度的循环码为LCD码的一个充要条件，证明了 $? 4$ 上LCD循环码的二元像是可逆码；构造了 $? 4$ 上长为 $2 m + 1$ 的LCD循环码，得到了参数较好的二元非线性可逆码.

关键词: 线性码, 线性互补对偶（LCD）码, 循环码, 二元像, 可逆码

Abstract:

Cyclic codes and linear complementary dual (LCD) codes are two important classes of linear codes which have been extensively used in data storage systems, communication systems and cryptography. In this paper, LCD cyclic codes over $? 4$ of odd length are explored. A sufficient and necessary condition is given for cyclic codes over $? 4$ of odd length to be LCD. It is proved that the binary images of LCD cyclic codes over $? 4$ are reversible codes. LCD cyclic codes over $? 4$ of length $2 m + 1$ are constructed. And binary nonlinear reversible codes with good parameters are obtained.

Key words: linear code, linear complementary dual (LCD) code, cyclic code, binary image, reversible code

中图分类号:

TN911.22

开晓山, 廖文敬. $ℤ 4$ 上LCD循环码及其二元像[J]. 电子学报, 2021, 49(11): 2284-2288.

KAI Xiao-shan, LIAO Wen-jing. LCD Cyclic Codes over ℤ₄ and Binary Images[J]. Acta Electronica Sinica, 2021, 49(11): 2284-2288.

参考文献 19

1	MasseyJ L. Linear codes with complementary duals[J]. Discrete Mathematics, 1992, 106(107): 337 - 342.
2	CarletC, GuilleyS. Complementary dual codes for counter-measures to side-channel attacks[A]. The 4th International Castle Meeting Coding Theory and Applications[C]. Palmela Castle, Portugal: Springer, 2015. 97 - 105.
3	BeelenP, JinL. Explicit MDS codes with complementary duals[J]. IEEE Transactions on Information Theory, 2018, 64(11): 7188 - 7193.
4	CarletC, MesnagerS, TangC, QiY. Euclidean and Hermitian LCD MDS codes[J]. Designs, Codes and Cryptography, 2018, 86(11): 2606 - 2618.
5	钱毅,李平,唐永生. 一种四元厄米特LCD码与厄米特自正交码的构造方法[J]. 电子学报,2020,48(3): 577 - 581.
	QianY, LiP, TangY S. A construction method of quaternary Hermitian LCD codes and Hermitian self-orthogonal codes[J]. Acta Electronica Sinica, 2020, 48(3): 577 - 581.(in Chinese)
6	CarletC, MesnagerS, TangC, et al. Linear codes over 𝔽q are equivalent to LCD codes for q>3[J]. IEEE Transactions on Information Theory, 2018, 64(4): 3010 - 3017.
7	LiuX, LiuH. LCD codes over ﬁnite chain rings[J]. Finite Fields and Their Applications, 2015, 34(7): 1 - 19.
8	ShiM, ZhuH, QianL, et al. On self-dual and LCD double circulant and double negacirculant codes over 𝔽q+u𝔽q[J]. Cryptography and Communications, 2020, 12(1): 53 - 70.
9	LiuZ, WangJ. Linear complementary dual codes over rings[J]. Designs, Codes and Cryptography, 2019, 87(11): 3077 - 3086.
10	AlahmadiA, AlhazmiH, HellesethT, et al. On the lifted Melas code[J]. Cryptography and Communications, 2016, 8(1): 7 - 18.
11	YangX, MasseyJ L. The necessary and sufficient condition for a cyclic code to have a complementary dual[J]. Discrete Mathematics, 1994, 126(3): 391 - 393.
12	LiC, DingC, LiS. LCD cyclic codes over ﬁnite ﬁelds[J]. IEEE Transactions on Information Theory, 2017, 63(7) : 4344 - 4356.
13	WanZ. Quaternary Codes[M]. Singapore: World Scientiﬁc, 1997.
14	NortonG H, SalageanA. On the Hamming distance of linear codes over a finite chain ring[J]. IEEE Transactions on Information Theory, 2000, 46(3): 1060 - 1067.
15	NortonG H, SalageanA. On the structure of linear and cyclic codes over a ﬁnite chain ring[J]. Applicable Algebra in Engineering, Communication and Computing, 2000, 10(7): 489 - 506.
16	WolfmannJ. Binary images of cyclic codes over ℤ4[J]. IEEE Transactions on Information Theory, 2001, 47(5): 1773 - 1779.
17	MacWilliamsF J, SloaneN J A. The Theory of Error-Correcting Codes[M]. Amsterdam, Netherland: North-Holland Publishing Company, 1977.
18	GrasslM. Bounds on the minimum distance of linear codes and quantum codes[N]. , 2020-06-15.
19	InterlandoJ C, PalazzoR, EliaM. On the decoding of Reed-Solomon and BCH codes over integer residue rings[J]. IEEE Transactions on Information Theory, 1997, 43(3): 1013 - 1021.

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$ℤ 4$ 上LCD循环码及其二元像

LCD Cyclic Codes over ℤ₄ and Binary Images

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