Table 1.
LCD negacyclic codes1
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Repeated-root constacyclic codes of length 6lmpn
| 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 |
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
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References:
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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. |
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E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill Book Company, New York, 1968. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003.
doi: 10.1142/5350. |
| [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.
|
| [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. |
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] |
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill Book Company, New York, 1968. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
Z. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing, Singapore, 2003.
doi: 10.1142/5350. |
| [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.
|
| [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. |
| 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 |
| 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 |
| 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 |
| 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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