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An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $
1. | School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China |
2. | Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China |
3. | School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China |
4. | Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam |
5. | Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam |
6. | Department of Mathematics, Dr. SPM IIIT Naya Raipur, Atal Nagar 493661, India |
7. | Chern Institute of Mathematics and LPMC, Nankai University, Tianjin Key Laboratory of Network and Data Security Technology, Tianjin 300071, China |
In this paper, we give an explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over the local non-principal ideal ring $ R = \mathbb{Z}_4+u\mathbb{Z}_4 $ $ (u^2 = 0) $, where $ k, n $ are arbitrary positive integers and $ n $ is odd. In particular, we present all distinct negacyclic codes of length $ 2^k $ over $ R $ precisely. Moreover, we provide an exact mass formula for the number of negacyclic codes of length $ 2^kn $ over $ R $ and correct several mistakes in some literatures.
References:
[1] |
T. Abualrub and R. Oehmke,
On the generators of $\mathbb{Z}_4$ cyclic codes of lenth $2^e$, IEEE Trans. Inform. Theory, 49 (2003), 2126-2133.
doi: 10.1109/TIT.2003.815763. |
[2] |
T. Abualrub and I. Siap,
Cyclic codes over the ring $\mathbb{Z}_2+u\mathbb{Z}_2$ and $\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.
doi: 10.1007/s10623-006-9034-5. |
[3] |
R. Bandi and M. Bhaintwal, Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, 2015, https://www.researchgate.net/publication/289506486. Google Scholar |
[4] |
R. Bandi, M. Bhaintwal and N. Aydin,
A mass formula for negacyclic codes of length $2^k$ and some good negacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 241-272.
doi: 10.1007/s12095-015-0172-3. |
[5] |
T. Blackford,
Negacyclic codes over $\mathbb{Z}_4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424.
doi: 10.1109/TIT.2003.811915. |
[6] |
Y. Cao,
On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.
doi: 10.1016/j.ffa.2013.07.001. |
[7] |
Y. Cao and Q. Li,
Cyclic codes of odd length over $\mathbb{Z}_4[u]/\langle u^k\rangle$, Cryptogr. Commun., 9 (2017), 599-624.
doi: 10.1007/s12095-016-0204-7. |
[8] |
Y. Cao, Y. Cao and F.-W. Fu,
Cyclic codes over $\mathbb{F}_{2^m}[u]/\langle u^k \rangle$ of oddly even length, Appl. Algebra in Engrg. Commun. Comput., 27 (2016), 259-277.
doi: 10.1007/s00200-015-0281-4. |
[9] |
Y. Cao, Y. Cao and Q. Li,
Concatenated structure of cyclic codes over $\mathbb{Z}_4$ of length $4n$, Appl. Algebra in Engrg. Commun. Comput., 27 (2016), 279-302.
doi: 10.1007/s00200-015-0283-2. |
[10] |
Y. Cao, Y. Cao, S. T. Dougherty and S. Ling,
Construction and enumeration for self-dual cyclic codes over $\mathbb{Z}_4$ of oddly even length, Des. Codes Cryptogr., 87 (2019), 2419-2446.
doi: 10.1007/s10623-019-00629-6. |
[11] |
Y. Cao, Y. Cao and Q. Li,
The concatenated structure of cyclic codes over $\mathbb{Z}_{p^2}$, J. Appl. Math. Comput., 52 (2016), 363-385.
doi: 10.1007/s12190-015-0945-z. |
[12] |
Y. Cao and Y. Cao,
Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images, Finite Fields Appl., 52 (2018), 67-93.
doi: 10.1016/j.ffa.2018.03.005. |
[13] |
Y. Cao and Y. Cao,
Complete classification for simple root cyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$, Cryptogr. Commun., (2019), 1-19.
doi: 10.1007/s12095-019-00403-4. |
[14] |
Y. Cao and Y. Cao, Complete classification for simple-root cyclic codes over $\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$, 2017, https://www.researchgate.net/publication/320620031. Google Scholar |
[15] |
Y. Cao, Y. Cao, H. Q. Dinh, F.-W. Fu, J. Gao and S. Sriboonchitta,
Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m} $, Adv. Math. Commun., 12 (2018), 231-262.
doi: 10.3934/amc.2018016. |
[16] |
Y. Cao, Y. Cao, R. Bandi and F.-W. Fu, An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb{Z}_4+u\mathbb{Z}_4$, arXiv: 1811.10991 Google Scholar |
[17] |
H. Q. Dinh and S. R. López-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
[18] |
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. |
[19] |
H. Q. Dinh, S. Dhompongsa and S. Sriboonchitta,
Repeated-root constacyclic codes of prime power length over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^a\rangle}$ and their duals, Discrete Math., 339 (2016), 1706-1715.
doi: 10.1016/j.disc.2016.01.020. |
[20] |
S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[21] |
S. T. Dougherty and S. Ling,
Cyclic codes over $\mathbb{Z}_4$ of even length, Des. Codes Cryptogr., 39 (2006), 127-153.
doi: 10.1007/s10623-005-2773-x. |
[22] |
G. Norton and A. Sǎlǎgean-Mandache,
On the structure of linear and cyclic codes over finite chain rings, Appl. Algebra in Engrg. Comm. Comput., 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
[23] |
P. Pattanayak and A. K. Singh, A class of cyclic codes cver the ring $\mathbb{Z}_4[u]/\langle u^2\rangle$ and its gray image, arXiv: 1507.04938 Google Scholar |
[24] |
M. Shi, L. Xu and G. Yang,
A note on one weight and two weight projective $\mathbb{Z}_4$-codes, IEEE Trans. Inform. Theory, 63 (2017), 177-182.
doi: 10.1109/TIT.2016.2628408. |
[25] |
M. Shi, L. Qian, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[26] |
Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/5350. |
[27] |
J. A. Wood,
Duality for modules over finite rings and applications to coding theory, American Journal of Mathematics, 121 (1999), 555-575.
doi: 10.1353/ajm.1999.0024. |
[28] |
B. Yildiz and S. Karadeniz,
Linear codes over $\mathbb{Z}_4+u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27 (2014), 24-40.
doi: 10.1016/j.ffa.2013.12.007. |
[29] |
B. Yildiz and N. Aydin,
Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ and $\mathbb{Z}_4$ images, International Journal of Information and Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
show all references
References:
[1] |
T. Abualrub and R. Oehmke,
On the generators of $\mathbb{Z}_4$ cyclic codes of lenth $2^e$, IEEE Trans. Inform. Theory, 49 (2003), 2126-2133.
doi: 10.1109/TIT.2003.815763. |
[2] |
T. Abualrub and I. Siap,
Cyclic codes over the ring $\mathbb{Z}_2+u\mathbb{Z}_2$ and $\mathbb{Z}_2+u\mathbb{Z}_2+u^2\mathbb{Z}_2$, Des. Codes Cryptogr., 42 (2007), 273-287.
doi: 10.1007/s10623-006-9034-5. |
[3] |
R. Bandi and M. Bhaintwal, Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, 2015, https://www.researchgate.net/publication/289506486. Google Scholar |
[4] |
R. Bandi, M. Bhaintwal and N. Aydin,
A mass formula for negacyclic codes of length $2^k$ and some good negacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 241-272.
doi: 10.1007/s12095-015-0172-3. |
[5] |
T. Blackford,
Negacyclic codes over $\mathbb{Z}_4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424.
doi: 10.1109/TIT.2003.811915. |
[6] |
Y. Cao,
On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.
doi: 10.1016/j.ffa.2013.07.001. |
[7] |
Y. Cao and Q. Li,
Cyclic codes of odd length over $\mathbb{Z}_4[u]/\langle u^k\rangle$, Cryptogr. Commun., 9 (2017), 599-624.
doi: 10.1007/s12095-016-0204-7. |
[8] |
Y. Cao, Y. Cao and F.-W. Fu,
Cyclic codes over $\mathbb{F}_{2^m}[u]/\langle u^k \rangle$ of oddly even length, Appl. Algebra in Engrg. Commun. Comput., 27 (2016), 259-277.
doi: 10.1007/s00200-015-0281-4. |
[9] |
Y. Cao, Y. Cao and Q. Li,
Concatenated structure of cyclic codes over $\mathbb{Z}_4$ of length $4n$, Appl. Algebra in Engrg. Commun. Comput., 27 (2016), 279-302.
doi: 10.1007/s00200-015-0283-2. |
[10] |
Y. Cao, Y. Cao, S. T. Dougherty and S. Ling,
Construction and enumeration for self-dual cyclic codes over $\mathbb{Z}_4$ of oddly even length, Des. Codes Cryptogr., 87 (2019), 2419-2446.
doi: 10.1007/s10623-019-00629-6. |
[11] |
Y. Cao, Y. Cao and Q. Li,
The concatenated structure of cyclic codes over $\mathbb{Z}_{p^2}$, J. Appl. Math. Comput., 52 (2016), 363-385.
doi: 10.1007/s12190-015-0945-z. |
[12] |
Y. Cao and Y. Cao,
Negacyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$ of oddly even length and their Gray images, Finite Fields Appl., 52 (2018), 67-93.
doi: 10.1016/j.ffa.2018.03.005. |
[13] |
Y. Cao and Y. Cao,
Complete classification for simple root cyclic codes over the local ring $\mathbb{Z}_4[v]/\langle v^2+2v\rangle$, Cryptogr. Commun., (2019), 1-19.
doi: 10.1007/s12095-019-00403-4. |
[14] |
Y. Cao and Y. Cao, Complete classification for simple-root cyclic codes over $\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$, 2017, https://www.researchgate.net/publication/320620031. Google Scholar |
[15] |
Y. Cao, Y. Cao, H. Q. Dinh, F.-W. Fu, J. Gao and S. Sriboonchitta,
Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m} + u\mathbb{F}_{p^m} $, Adv. Math. Commun., 12 (2018), 231-262.
doi: 10.3934/amc.2018016. |
[16] |
Y. Cao, Y. Cao, R. Bandi and F.-W. Fu, An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb{Z}_4+u\mathbb{Z}_4$, arXiv: 1811.10991 Google Scholar |
[17] |
H. Q. Dinh and S. R. López-Permouth,
Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.
doi: 10.1109/TIT.2004.831789. |
[18] |
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. |
[19] |
H. Q. Dinh, S. Dhompongsa and S. Sriboonchitta,
Repeated-root constacyclic codes of prime power length over $\frac{\mathbb{F}_{p^m}[u]}{\langle u^a\rangle}$ and their duals, Discrete Math., 339 (2016), 1706-1715.
doi: 10.1016/j.disc.2016.01.020. |
[20] |
S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu,
Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 14-26.
doi: 10.1016/j.ffa.2009.11.004. |
[21] |
S. T. Dougherty and S. Ling,
Cyclic codes over $\mathbb{Z}_4$ of even length, Des. Codes Cryptogr., 39 (2006), 127-153.
doi: 10.1007/s10623-005-2773-x. |
[22] |
G. Norton and A. Sǎlǎgean-Mandache,
On the structure of linear and cyclic codes over finite chain rings, Appl. Algebra in Engrg. Comm. Comput., 10 (2000), 489-506.
doi: 10.1007/PL00012382. |
[23] |
P. Pattanayak and A. K. Singh, A class of cyclic codes cver the ring $\mathbb{Z}_4[u]/\langle u^2\rangle$ and its gray image, arXiv: 1507.04938 Google Scholar |
[24] |
M. Shi, L. Xu and G. Yang,
A note on one weight and two weight projective $\mathbb{Z}_4$-codes, IEEE Trans. Inform. Theory, 63 (2017), 177-182.
doi: 10.1109/TIT.2016.2628408. |
[25] |
M. Shi, L. Qian, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[26] |
Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/5350. |
[27] |
J. A. Wood,
Duality for modules over finite rings and applications to coding theory, American Journal of Mathematics, 121 (1999), 555-575.
doi: 10.1353/ajm.1999.0024. |
[28] |
B. Yildiz and S. Karadeniz,
Linear codes over $\mathbb{Z}_4+u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self-dual codes, Finite Fields Appl., 27 (2014), 24-40.
doi: 10.1016/j.ffa.2013.12.007. |
[29] |
B. Yildiz and N. Aydin,
Cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ and $\mathbb{Z}_4$ images, International Journal of Information and Coding Theory, 2 (2014), 226-237.
doi: 10.1504/IJICOT.2014.066107. |
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