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  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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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