# American Institute of Mathematical Sciences

May  2021, 15(2): 291-309. doi: 10.3934/amc.2020067

## 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

* Corresponding author: Yonglin Cao

Received  June 2019 Revised  October 2019 Published  May 2021 Early access  January 2020

Fund Project: This research is supported in part by National Natural Science Foundation of China (Grant Nos. 11801324, 11671235, 61971243, 61571243), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Foundation for the PhD of Shandong University of Technology (Grant No. 417037), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University) (Grant Nos. HBAM201906, HBAM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09) and the Nankai Zhide Foundation

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.

Citation: Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb{Z}_4+u\mathbb{Z}_4$. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067
##### 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. [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. [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 [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 [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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##### 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. [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. [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 [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 [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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