$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})-$ linear skew constacyclic codes

Yıl 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 85 - 101, 29.02.2020

Ahlem Melakhessou Nuh Aydin , Zineb Hebbache Kenza Guenda

https://doi.org/10.13069/jacodesmath.671815

Cited By: 8

Öz

In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\Theta]$ are discussed, where $ \Theta$ is an automorphism of $R.$ We describe the generator polynomials of skew constacyclic codes over $\mathbb{Z}_{q}R,$ also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_{4}$. Finally, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.

Anahtar Kelimeler

Linear codes , Skew constacyclic codes , $\mathbb{Z}_{q}\mathbb{Z}_{q} , Bounds

Kaynakça

• [1] T. Abualrub, I. Siap, Cyclic codes over the rings Z2 +uZ2 and Z2 +uZ2 +u2Z2, Designs, Codes and Cryptography, 42 (3), pp. 273–287, 2007.
• [2] T. Abualrub, I. Siap and I. Aydogdu, Z2(Z2 + uZ2)-Linear cyclic codes, Proceedings of the IMECS 2014, (2), Hong Kong, 2014.
• [3] T. Abualrub, I. Siap, and N. Aydin, Z2Z4􀀀additive cyclic codes, IEEE. Trans. Inf. Theory, vol. 60, no. 3, pp. 1508–514, 2014.
• [4] R. Ackerman and N. Aydin, New quinary linear codes from quasi-twisted codes and their duals, Appl. Math. Lett., 24(4), pp. 512–515, 2011.
• [5] J. B. Ayats, C. F. Córdoba and R. T. Valls, Z2Z4-additive cyclic codes, generator polynomials and dual codes, IEEE Transactions on Information Theory, (62), pp. 6348–6354, 2016.
• [6] I. Aydogdu, T. Abualrub and I. Siap, Z2Z2[u]􀀀cyclic and constacyclic codes, IEEE Transactions on Information Theory, 63 (8), pp. 4883–4893, 2016.
• [7] N. Aydin and T. Asamov, A Database of Z4 Codes, Journal of Combinatorics, Information & System Sciences, 34 (1-4), pp. 1–12, 2009.
• [8] N. Aydin, N. Connolly and M. Grassl, Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes, Adv. Math. of Commun., 11 (1), pp. 245–258, 2017.
• [9] N. Aydin, N. Connolly and J. Murphree, New binary linear codes from QC codes and an augmentation algorithm, Appl. Algebra Eng. Commun. Comput., 28( 4), pp. 339–350, 2017.
• [10] N. Aydin, Y. Cengellenmis and A. Dertli, On some constacyclic codes over Z4[u]=hu2 􀀀 1i, their Z4 images, and new codes, Designs, Codes and Cryptography, 86 (6), pp. 1249–1255, 2018.
• [11] N. Aydin, I. Siap and D. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Designs, Codes and Cryptography, 24 (3), pp. 313–326, 2001.
• [12] N. Aydin and I. Siap, New quasi-cyclic codes over F5, Appl. Math. Lett., 15 (7), pp. 833–836, 2002. [13] N. Aydin and A. Halilovic, A Generalization of Quasi-twisted Codes: Multi-twisted codes, Finite Fields and Their Applications, (45 ), pp. 96–106, 2017.
• [14] R. K. Bandi and M. Bhaintwal, A note on cyclic codes over Z4 + uZ4, Discrete Mathematics, Algorithms and Applications, 8 (1), pp. 1–17, 2016.
• [15] N. Bennenni, K. Guenda and S. Mesnager, DNA cyclic codes over rings, Adv. in Math. of Comm., 11 (1), pp. 83–98, 2017.
• [16] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18(4), pp. 379–389, 2007.
• [17] R. Daskalov, P. Hristov, New binary one-generator quasi-cyclic codes, IEEE Trans. Inf. Theory, 49 (11), pp 3001–3005, 2003.
• [18] R. Daskalov, P. Hristov and E. Metodieva, New minimum distance bounds for linear codes over GF(5), Discrete Math., 275 (1–3), pp. 97–110, 2004.
• [19] Database of Z4 Codes. [online] Z4Codes. info (Accessed March, 2018).
• [20] H. Q. Dinh, A. K. Singh, S. Pattanayak and S. Sriboonchitta, Cyclic DNA codes over the ringF2 + uF2 + vF2 + uvF2 + v2F2 + uv2F2, Designs, Codes and Cryptography, 86 (7), pp. 1451–1467,2018.
• [21] M.F. Ezerman, S. Ling, P. Solé and O. Yemen, From skew-cyclic codes to asymmetric quantum code,Adv. in Math. of Comm., 5 (1), pp. 41–57, 2011.
• [22] J. Gao., Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform., 31 (3–4), pp. 337–342, 2013.
• [23] I. Siap and N. Kulhan, The Structure of Generalized Quasi Cyclic Codes, Appl. Math. E-Notes, vol. 5, pp. 24–30, 2005.
• [24] J. Gao, F. W. Fu, L. Xiao and R. K. Bandi, Some results on cyclic codes over Zq + uZq, Discrete Mathematics, Algorithms and Applications, 7 (4), pp. 1–9, 2015.
• [25] J. Gao, F. Ma and F. Fu, Skew constacyclic codes over the ring Fq + vFq; Appl.Comput. Math., 6 (3), pp. 286–295, 2017 .
• [26] M. Grassl, Code Tables: Bounds on the parameters of codes, online, http://www.codetables.de/
• [27] F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over Fq + vFq, Advances in Mathematics of Communications, 8 (3), pp. 313–322, 2014.
• [28] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic over finite chain rings, Adv. Math.Commun., 6 (1), pp. 39–63, 2012.
• [29] P. Li, W. Dai and X. Kai, On Z2Z2[u]􀀀(1+u)-additive constacyclic, arXiv:1611.03169v1 [cs.IT] 10 Nov 2016.
• [30] Magma computer algebra system, online, http://magma.maths.usyd.edu.au/
• [31] J. F. Qian, L. N. Zhang and S. X. Zhu, (1+u)-Constacyclic and cyclic codes over F2 +uF2, Applied Mathematics Letters, 19 (8), pp. 820–823, 2006.
• [32] A. Sharma and M. Bhaintwal, A class of skew-constacyclic codes over Z4 + uZ4, Int. J. Information and Coding Theory, 4 (4), pp. 289–303, 2017.
• [33] I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Information and Coding Theory, 2 (1), pp. 10–20, 2011.
• [34] B. Yildiz, N. Aydin, Cyclic codes over Z4 +uZ4 and their Z4-images , Int. J. Information and coding Theory, 2 (4), pp. 226–237, 2014.