Repeated-root constacyclic codes of length 6lmpn

Tingting Wu; Shixin Zhu; Li Liu; Lanqiang Li

doi:10.3934/amc.2021044

Article Contents

2023, Volume 17, Issue 5: 1154-1180. Doi: 10.3934/amc.2021044

This issue Previous Article Optimal quinary negacyclic codes with minimum distance four Next Article On BCH split metacyclic codes

Repeated-root constacyclic codes of length 6l^mpⁿ

1.
School of Mathematics, Hefei University of Technology, Hefei 230009, China
2.
Intelligent Interconnected Systems Laboratory of Anhui Province Hefei, China
3.
School of Science, Anhui Agricultural University, Hefei 230036, Anhui, China

^* Corresponding author: Li Liu
^* Corresponding author: Li Liu

Received: March 2021

Early access: September 2021

Published: October 2023

This research is supported in part by the National Natural Science Foundation of China under Project 61772168, 12001002, the fundamental Research Funds for the Central Universities under Project PA2021KCPY0040 and the Natural Science Foundation of Anhui Province under Project 2108085QA06, 2008085QA04, 2108085QA03

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Abstract

Let $ \mathbb{F}_{q} $ be a finite field with character $ p $. In this paper, the multiplicative group $ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $ is decomposed into a mutually disjoint union of $ \gcd(6l^mp^n,q-1) $ cosets over subgroup $ <\xi^{6l^mp^n}> $, where $ \xi $ is a primitive element of $ \mathbb{F}_{q} $. Based on the decomposition, the structure of constacyclic codes of length $ 6l^mp^n $ over finite field $ \mathbb{F}_{q} $ and their duals is established in terms of their generator polynomials, where $ p\neq{3} $ and $ l\neq{3} $ are distinct odd primes, $ m $ and $ n $ are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length $ 6l^mp^n $ over $ \mathbb{F}_{q} $.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. LCD negacyclic codes¹

q	l	m	length	dimension	minimum distance
5	7	2	1470	1460	2
5	7	2	1470	1050	6
5	7	2	1470	60	84
5	17	1	510	190	17
5	19	1	570	110	38
13	5	2	1950	1404	5
13	5	2	1950	858	10
13	5	2	1950	26	125
25	11	1	330	320	2
25	11	1	330	260	6
25	11	1	330	120	14

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Table 2. LCD negacyclic codes²

q	l	length	dimension	minimum distance
7	11	462	224	13
7	11	462	154	18
7	13	546	364	6
7	13	546	196	17
7	13	546	168	22
7	31	1302	1288	2
7	31	1302	14	93
11	19	1254	1078	6
11	19	1254	308	24
11	19	1254	154	55

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Table 3. self-dual negacyclic codes

q	l	m	length	dimension	minimum distance
13	5	2	1950	975	6
5	7	2	1470	735	8
5	11	1	330	165	8
5	17	1	510	255	8
5	19	1	570	285	8
5	23	1	690	345	8

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References

[1]	G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377. doi: 10.1016/j.ffa.2011.09.005.
[2]	E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill Book Company, New York, 1968.
[3]	B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length lp^s and their duals, Discrete Appl. Math., 177 (2014), 60-70. doi: 10.1016/j.dam.2014.05.046.
[4]	B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length 2l^mpⁿ, Finite Fields Appl., 33 (2015), 137-159. doi: 10.1016/j.ffa.2014.11.006.
[5]	B. Chen, H. Liu and G. Zhang, A class of minimal cyclic codes over finite fields, Des. Codes Cryptogr., 74 (2015), 285-300. doi: 10.1007/s10623-013-9857-9.
[6]	H. Q. Dinh, Repeated-root constacyclic codes of length 2p^s, Finite Fields Appl., 18 (2012), 133-143. doi: 10.1016/j.ffa.2011.07.003.
[7]	H. Q. Dinh, Structure of repeated-root constacyclic codes of length 3p^s and their duals, Discrete Math., 313 (2013), 983-991. doi: 10.1016/j.disc.2013.01.024.
[8]	H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length 6p^s and their duals, Contemp. Math., 609 (2014), 69-87. doi: 10.1090/conm/609/12150.
[9]	H. Q. Dinh and Sa roj Rani, Structure of some classes of repeated-root constacyclic codes of length 2^kl^mpⁿ, Discrete Math., 342 (2019), 111609. doi: 10.1016/j.disc.2019.111609.
[10]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[11]	L. Liu, L. Li, X. Kai and S. Zhu, Repeated-root constacyclic codes of length 3lp^s and their dual codes, Finite Fields Appl., 42 (2016), 269-295. doi: 10.1016/j.ffa.2016.08.005.
[12]	Y. Liu, M. Shi, H. Q. Dinh and S. Sriboonchitta, Repeated-root constacyclic codes of length 3l^mp^s, Advances in Math. Comm., 14 (2020), 359-378. doi: 10.1017/CBO9780511807077.
[13]	S. Rani, Structure of repeated-root constacyclic codes of length 8l^mpⁿ, Asian-Eur. J. Math., 12(2019), 1950050, 17 pp. doi: 10.1016/j.ffa.2016.08.005.
[14]	A. Sharma, Self-dual and self-orthogonal negacyclic codes of length 2^mpⁿ over a finite field, Discrete Math., 338 (2015), 576-592. doi: 10.1016/j.disc.2014.11.008.
[15]	A. Sharma, Repeated-root constacyclic codes of length l^tp^^s and their dual codes, Cryptogr. Commun., 7 (2015), 229-255. doi: 10.1007/s12095-014-0106-5.
[16]	A. Sharma and S. Rani, Repeated-root constacyclic codes of length 4l^mpⁿ, Finite Fields Appl., 40 (2016), 163-200. doi: 10.1016/j.ffa.2016.04.001.
[17]	Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003. doi: 10.1142/5350.
[18]	T. Wu, L. Liu, L. Li and S. Zhu, Repeated-root constacyclic codes of length 6lp^s and their dual codes, Advances in Math., 15 (2021), 167-189.
[19]	W. Zhao, X. Tang and Z. Gu, Constacyclic codes of length kl^mpⁿ over a finite field, Finite Fields Appl., 52 (2018), 51-66. doi: 10.1016/j.ffa.2018.03.004.