Repeated-root constacyclic codes of length $ 6lp^s $

Tingting Wu; Li Liu; Lanqiang Li; Shixin Zhu

doi:10.3934/amc.2020051

Article Contents

2021, Volume 15, Issue 1: 167-189. Doi: 10.3934/amc.2020051

This issue Previous Article Some properties of the cycle decomposition of WG-NLFSR Next Article Properties of sets of subspaces with constant intersection dimension

Repeated-root constacyclic codes of length $ 6lp^s $

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

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

Received: March 31, 2019

Revised: August 31, 2019

Early access: January 2020

Published: February 2021

This research is supported in part by the National Natural Science Foundation of China under Project 11871187 and Project 61772168

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Abstract

Let $ \mathbb{F}_{q} $ be a finite field with character $ p $ and $ p\neq{3},l\neq{3} $ be different odd primes. In this paper, we study constacyclic codes of length $ 6lp^s $ over finite field $ \mathbb{F}_{q} $. The generator polynomials of all constacyclic codes and their duals are obtained. Moreover, we give the characterization and enumeration of linear complementary dual (LCD) and self-dual constacyclic codes of length $ 6lp^s $ over $ \mathbb{F}_{q} $.

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Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.

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Table 1. LCD cyclic codes

$ p $	l	length	dimension	minimum distance
$ 5 $	$ 29 $	$ 870 $	$ 785 $	3
$ 5 $	$ 29 $	$ 870 $	$ 780 $	4
$ 5 $	$ 29 $	$ 870 $	$ 10 $	116
$ 5 $	$ 29 $	$ 870 $	$ 5 $	174
$ 11 $	$ 7 $	$ 462 $	$ 374 $	4
$ 11 $	$ 7 $	$ 462 $	$ 242 $	6
$ 11 $	$ 7 $	$ 462 $	$ 165 $	8

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Table 2. LCD negacyclic codes

$ p $	l	length	dimension	minimum distance
$ 11 $	5	$ 330 $	$ 286 $	3
$ 11 $	5	$ 330 $	$ 264 $	4
$ 11 $	5	$ 330 $	$ 242 $	5
$ 11 $	5	$ 330 $	$ 176 $	7
$ 11 $	5	$ 330 $	$ 154 $	8
$ 13 $	11	$ 858 $	$ 594 $	3
$ 13 $	11	$ 858 $	$ 550 $	4
$ 37 $	5	$ 1110 $	$ 814 $	3
$ 37 $	5	$ 1110 $	$ 666 $	5
$ 37 $	5	$ 1110 $	$ 592 $	6
$ 37 $	5	$ 1110 $	$ 296 $	10
$ 37 $	5	$ 1110 $	$ 148 $	12

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Table 3. self-dual negacyclic codes

$ p $	l	length	dimension	minimum distance
$ 13 $	5	$ 390 $	$ 195 $	6
$ 13 $	11	$ 858 $	$ 429 $	6
$ 13 $	19	$ 1482 $	$ 741 $	6
$ 37 $	5	$ 1110 $	$ 555 $	6

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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]	G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 337-342. doi: 10.1109/18.75249.
[3]	B. C. Chen, H. Q. Dinh and H. W. Liu, Repeated-root constacyclic codes of length $lp^s$ and their duals, Discrete Math., 177 (2014), 60-70. doi: 10.1016/j.dam.2014.05.046.
[4]	B. C. Chen, H. Q. Dinh and H. W. Liu, Repeated-root constacyclic codes of length $2l^mp^n$, Finite Fields Appl., 33 (2015), 137-159. doi: 10.1016/j.ffa.2014.11.006.
[5]	H. Q. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions, Finite Fields Appl., 14 (2008), 22-40. doi: 10.1016/j.ffa.2007.07.001.
[6]	H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+ u\mathbb {F}_{{p^m}}$, J. Algebra, 324 (2010), 940-950. doi: 10.1016/j.jalgebra.2010.05.027.
[7]	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.
[8]	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.
[9]	H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length $6p^s$ and their duals, Contemp. Math., 609 (2014), 69-87.
[10]	Y. Jia, S. Ling and C. P. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inf. Theory, 57 (2011), 2243-2251. doi: 10.1109/TIT.2010.2092415.
[11]	X. S. Kai and S. X. Zhu, On the distance of cyclic codes of length $2^e$ over $Z_{4}$, Discrete Math., 310 (2010), 12-20. doi: 10.1016/j.disc.2009.07.018.
[12]	L. Katburia and M. Raka, Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless, Adv. Math. Commun, 6 (2012), 499-503. doi: 10.3934/amc.2012.6.499.
[13]	L. Liu, L. Q. Li, X. S. Kai and S. X. 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.
[14]	A. Sharma, G. K. Bakshi, V. C. Dumir and M. Raka, Cyclotomic numbers and primitive idempotents in the ring $GF(q)[x]/ \langle {x^{p^{n}}-1} \rangle$, Finite Fields Appl., 10 (2004), 653-673. doi: 10.1016/j.ffa.2004.01.005.
[15]	H. X. Tong, Repeated-root constacyclic codes of length $kl^ap^b$ over a finite field, Finite Fields Appl., 41 (2016), 159-173. doi: 10.1016/j.ffa.2016.06.006.
[16]	J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 343-345. doi: 10.1109/18.75250.
[17]	Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.
[18]	W. Zhao, X. L. Tang and Z. Gu, Constacyclic codes of length $kl^mp^n$ over a finite field, Finite Fields Appl., 52 (2018), 51-66. doi: 10.1016/j.ffa.2018.03.004.