American Institute of Mathematical Sciences

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May  2020, 14(2): 359-378. doi: 10.3934/amc.2020025

Repeated-root constacyclic codes of length $ 3\ell^mp^s $

Yan Liu 1, , Minjia Shi 1,, , Hai Q. Dinh 2,3, and  Songsak Sriboonchitta 4,

1. 

Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China
2. 

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3. 

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4. 

Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai 52000, Thailand

* Corresponding author

Received  August 2018 Revised  May 2016 Published  May 2020 Early access  September 2019

Fund Project: This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20) and Academic fund for outstanding talents in universities (gxbjZD03). H. Q. Dinh and S. Sriboonchitta are grateful to the Centre of Excellence in Econometrics, Chiang Mai University, for partial financial support. This research is partially supported by the Research Administration Center, Chaing Mai University

Let $ p $ be a prime different from 3, and $ \ell $ be an odd prime different from 3 and $ p $. In terms of generator polynomials, structures of constacyclic codes and their duals of length $ 3\ell^mp^s $ over $ \mathbb{F}_q $ are established, where $ q $ is a power of $ p $. We discuss the enumeration of all cyclic codes of length $ 3\cdot2^s\ell^m $, that generalizes the construction of [15] (2016), which is the special case of $ m = 1 $. In addition, as an application, the characterization and enumeration of all linear complementary dual cyclic codes of length $ 6\ell^mp^s $ over $ \mathbb{F}_q $ are obtained.

Keywords: Constacyclic codes, Dual codes, generator polynomials, LCD cyclic codes.
Mathematics Subject Classification: Primary: 94B25; Secondary: 05E30.
Citation: Yan Liu, Minjia Shi, Hai Q. Dinh, Songsak Sriboonchitta. Repeated-root constacyclic codes of length $ 3\ell^mp^s $. Advances in Mathematics of Communications, 2020, 14 (2) : 359-378. doi: 10.3934/amc.2020025
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. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field, Finite Fields Appl., 19 (2013), 39-54.  doi: 10.1016/j.ffa.2012.10.003.

[3]

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.

[4]

B. C. Chen, H. Q. Dinh and H. W. Liu, Repeated-root constacyclic codes of length $\ell p^s$ and their duals, Discrete Appl. Math., 177 (2014), 60-70. 

[5]

H. Q. Dinh, Repeated-root constacyclic codes of length $2\ell^m p^n$, Finite Fields Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.

[6]

B. C. Chen, H. W. Liu and G. H. Zhang, A class of minimal cyclic codes over finite fields, Des. Codes Cryptogr., 74 (2015), 285-300.  doi: 10.1007/s10623-013-9857-9.

[7]

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.

[8]

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.

[9]

H. Q. Dinh, Repeated-root constacyclic codes of length $2 p^s$, Finite Fields Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.

[10]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3 p^s$ and their duals, Discrete Math., 313 (2013), 982-991.  doi: 10.1016/j.disc.2013.01.024.

[11]

H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length $6 p^s$ and their duals, Contemp. Math., Amer. Math. Soc., Providence, RI, 609 (2014), 69-87.  doi: 10.1090/conm/609/12150.

[12]

H. Q. Dinh, X. Wang, H. Liu and S. Sriboonchitta, Hamming distance of constacyclic codes of length $3p^s$ and optimal codes with respect to the Griesmer and Singleton bound, preprint, 2018.

[13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[14]

X. S. Kai and S. X. Zhu, On the distance of cyclic codes length $2^e$ over $ \mathbb{Z}_4$, Discrete Math., 310 (2010), 12-20.  doi: 10.1016/j.disc.2009.07.018.

[15]

L. Liu, L. Q. Li, X. S. Kai and S. X. Zhu, Repeated-root constacyclic codes of length $3\ell p^s$ and their dual codes, Finite Fields Appl., 42 (2016), 269-295.  doi: 10.1016/j.ffa.2016.08.005.

[16]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.

[17]

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.

[18]

H. X. Tong, Repeated-root constacyclic codes of length $k\ell^ap^b$ over a finite field, Finite Fields Appl., 41 (2016), 159-173.  doi: 10.1016/j.ffa.2016.06.006.

[19]

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

[20]

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

[21]

S. D. Yang, X. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields and Their Applications, 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[22]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight enumerator of the duals of a class of cyclic codes with three zeros, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 347-367.  doi: 10.1007/s00200-015-0255-6.

[23]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[24]

S. D. Yang and Z. A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

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.

[2]

G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field, Finite Fields Appl., 19 (2013), 39-54.  doi: 10.1016/j.ffa.2012.10.003.

[3]

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.

[4]

B. C. Chen, H. Q. Dinh and H. W. Liu, Repeated-root constacyclic codes of length $\ell p^s$ and their duals, Discrete Appl. Math., 177 (2014), 60-70. 

[5]

H. Q. Dinh, Repeated-root constacyclic codes of length $2\ell^m p^n$, Finite Fields Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.

[6]

B. C. Chen, H. W. Liu and G. H. Zhang, A class of minimal cyclic codes over finite fields, Des. Codes Cryptogr., 74 (2015), 285-300.  doi: 10.1007/s10623-013-9857-9.

[7]

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.

[8]

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.

[9]

H. Q. Dinh, Repeated-root constacyclic codes of length $2 p^s$, Finite Fields Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.

[10]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3 p^s$ and their duals, Discrete Math., 313 (2013), 982-991.  doi: 10.1016/j.disc.2013.01.024.

[11]

H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length $6 p^s$ and their duals, Contemp. Math., Amer. Math. Soc., Providence, RI, 609 (2014), 69-87.  doi: 10.1090/conm/609/12150.

[12]

H. Q. Dinh, X. Wang, H. Liu and S. Sriboonchitta, Hamming distance of constacyclic codes of length $3p^s$ and optimal codes with respect to the Griesmer and Singleton bound, preprint, 2018.

[13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[14]

X. S. Kai and S. X. Zhu, On the distance of cyclic codes length $2^e$ over $ \mathbb{Z}_4$, Discrete Math., 310 (2010), 12-20.  doi: 10.1016/j.disc.2009.07.018.

[15]

L. Liu, L. Q. Li, X. S. Kai and S. X. Zhu, Repeated-root constacyclic codes of length $3\ell p^s$ and their dual codes, Finite Fields Appl., 42 (2016), 269-295.  doi: 10.1016/j.ffa.2016.08.005.

[16]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.

[17]

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.

[18]

H. X. Tong, Repeated-root constacyclic codes of length $k\ell^ap^b$ over a finite field, Finite Fields Appl., 41 (2016), 159-173.  doi: 10.1016/j.ffa.2016.06.006.

[19]

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

[20]

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

[21]

S. D. Yang, X. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields and Their Applications, 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[22]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight enumerator of the duals of a class of cyclic codes with three zeros, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 347-367.  doi: 10.1007/s00200-015-0255-6.

[23]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[24]

S. D. Yang and Z. A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

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