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