doi: 10.3934/amc.2020051

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

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

* Corresponding author: Li Liu

Received  April 2019 Revised  September 2019 Published  January 2020

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

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} $.

Citation: Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020051
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.  Google Scholar

[2]

G. CastagnoliJ. L. MasseyP. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.  Google Scholar

[3]

B. C. ChenH. 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.  Google Scholar

[4]

B. C. ChenH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

Y. JiaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. LiuL. Q. LiX. 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.  Google Scholar

[14]

A. SharmaG. K. BakshiV. 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.  Google Scholar

[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.  Google Scholar

[16]

J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 343-345.  doi: 10.1109/18.75250.  Google Scholar

[17]

Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.  Google Scholar

[18]

W. ZhaoX. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

G. CastagnoliJ. L. MasseyP. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.  Google Scholar

[3]

B. C. ChenH. 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.  Google Scholar

[4]

B. C. ChenH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

Y. JiaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. LiuL. Q. LiX. 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.  Google Scholar

[14]

A. SharmaG. K. BakshiV. 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.  Google Scholar

[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.  Google Scholar

[16]

J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inf. Theory, 37 (1991), 343-345.  doi: 10.1109/18.75250.  Google Scholar

[17]

Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.  Google Scholar

[18]

W. ZhaoX. 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.  Google Scholar

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
$ 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
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
$ 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
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
$ 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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