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doi: 10.3934/amc.2021044
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Repeated-root constacyclic codes of length 6lmpn

Tingting Wu 1,2, , Shixin Zhu 1,2, , Li Liu 1,2,, and  Lanqiang Li 3,

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

Received  March 2021 Early access September 2021

Fund Project: 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

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} $.

Keywords: Constacyclic codes, dual codes, generator polynomials, LCD negacyclic codes, self-dual constacyclic codes.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Tingting Wu, Shixin Zhu, Li Liu, Lanqiang Li. Repeated-root constacyclic codes of length 6lmpn. Advances in Mathematics of Communications, doi: 10.3934/amc.2021044
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

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B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length 2lmpn, Finite Fields Appl., 33 (2015), 137-159.  doi: 10.1016/j.ffa.2014.11.006.  Google Scholar

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

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H. Q. Dinh and Sa roj Rani, Structure of some classes of repeated-root constacyclic codes of length 2klmpn, Discrete Math., 342 (2019), 111609.  doi: 10.1016/j.disc.2019.111609.  Google Scholar

[10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[11]

L. Liu, L. Li, X. Kai and S. Zhu, Repeated-root constacyclic codes of length 3lps and their dual codes, Finite Fields Appl., 42 (2016), 269-295.  doi: 10.1016/j.ffa.2016.08.005.  Google Scholar

[12]

Y. Liu, M. Shi, H. Q. Dinh and S. Sriboonchitta, Repeated-root constacyclic codes of length 3lmps, Advances in Math. Comm., 14 (2020), 359-378.  doi: 10.1017/CBO9780511807077.  Google Scholar

[13]

S. Rani, Structure of repeated-root constacyclic codes of length 8lmpn, Asian-Eur. J. Math., 12(2019), 1950050, 17 pp. doi: 10.1016/j.ffa.2016.08.005.  Google Scholar

[14]

A. Sharma, Self-dual and self-orthogonal negacyclic codes of length 2mpn over a finite field, Discrete Math., 338 (2015), 576-592.  doi: 10.1016/j.disc.2014.11.008.  Google Scholar

[15]

A. Sharma, Repeated-root constacyclic codes of length ltp^s and their dual codes, Cryptogr. Commun., 7 (2015), 229-255.  doi: 10.1007/s12095-014-0106-5.  Google Scholar

[16]

A. Sharma and S. Rani, Repeated-root constacyclic codes of length 4lmpn, Finite Fields Appl., 40 (2016), 163-200.  doi: 10.1016/j.ffa.2016.04.001.  Google Scholar

[17]

Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003. doi: 10.1142/5350.  Google Scholar

[18]

T. Wu, L. Liu, L. Li and S. Zhu, Repeated-root constacyclic codes of length 6lps and their dual codes, Advances in Math., 15 (2021), 167-189.   Google Scholar

[19]

W. Zhao, X. Tang and Z. Gu, Constacyclic codes of length klmpn 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]

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill Book Company, New York, 1968.  Google Scholar

[3]

B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length lps and their duals, Discrete Appl. Math., 177 (2014), 60-70.  doi: 10.1016/j.dam.2014.05.046.  Google Scholar

[4]

B. Chen, H. Q. Dinh and H. Liu, Repeated-root constacyclic codes of length 2lmpn, Finite Fields Appl., 33 (2015), 137-159.  doi: 10.1016/j.ffa.2014.11.006.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length 3ps and their duals, Discrete Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.  Google Scholar

[8]

H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length 6ps and their duals, Contemp. Math., 609 (2014), 69-87.  doi: 10.1090/conm/609/12150.  Google Scholar

[9]

H. Q. Dinh and Sa roj Rani, Structure of some classes of repeated-root constacyclic codes of length 2klmpn, Discrete Math., 342 (2019), 111609.  doi: 10.1016/j.disc.2019.111609.  Google Scholar

[10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[11]

L. Liu, L. Li, X. Kai and S. Zhu, Repeated-root constacyclic codes of length 3lps and their dual codes, Finite Fields Appl., 42 (2016), 269-295.  doi: 10.1016/j.ffa.2016.08.005.  Google Scholar

[12]

Y. Liu, M. Shi, H. Q. Dinh and S. Sriboonchitta, Repeated-root constacyclic codes of length 3lmps, Advances in Math. Comm., 14 (2020), 359-378.  doi: 10.1017/CBO9780511807077.  Google Scholar

[13]

S. Rani, Structure of repeated-root constacyclic codes of length 8lmpn, Asian-Eur. J. Math., 12(2019), 1950050, 17 pp. doi: 10.1016/j.ffa.2016.08.005.  Google Scholar

[14]

A. Sharma, Self-dual and self-orthogonal negacyclic codes of length 2mpn over a finite field, Discrete Math., 338 (2015), 576-592.  doi: 10.1016/j.disc.2014.11.008.  Google Scholar

[15]

A. Sharma, Repeated-root constacyclic codes of length ltp^s and their dual codes, Cryptogr. Commun., 7 (2015), 229-255.  doi: 10.1007/s12095-014-0106-5.  Google Scholar

[16]

A. Sharma and S. Rani, Repeated-root constacyclic codes of length 4lmpn, Finite Fields Appl., 40 (2016), 163-200.  doi: 10.1016/j.ffa.2016.04.001.  Google Scholar

[17]

Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003. doi: 10.1142/5350.  Google Scholar

[18]

T. Wu, L. Liu, L. Li and S. Zhu, Repeated-root constacyclic codes of length 6lps and their dual codes, Advances in Math., 15 (2021), 167-189.   Google Scholar

[19]

W. Zhao, X. Tang and Z. Gu, Constacyclic codes of length klmpn over a finite field, Finite Fields Appl., 52 (2018), 51-66.  doi: 10.1016/j.ffa.2018.03.004.  Google Scholar

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
Table Options

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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
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
Table Options

Download as excel

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
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
Table Options

Download as excel

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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