Z_q(Z_q+uZ_q)-Linear Skew Constacyclic Codes
Keywords:
Linear codes, Skew constacyclic codes, ZqZq[u]− linear skew constacyclic codes, Bounds
Abstract
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 $\mathbb{Z}_{q}R$.
References
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.
T. Abualrub, I. Siap and I. Aydogdu, Z2(Z2 + uZ2)-Linear cyclic codes, Proceedings of the IMECS 2014, (2), Hong Kong, 2014.
T. Abualrub, I. Siap, and N. Aydin, Z2Z4
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.
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.
I. Aydogdu, T. Abualrub and I. Siap, Z2Z2[u]
N. Aydin and T. Asamov, A Database of Z4 Codes, Journal of Combinatorics, Information & System Sciences, 34 (1-4), pp. 1–12, 2009.
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.
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.
N. Aydin, Y. Cengellenmis and A. Dertli, On some constacyclic codes over Z4[u]=hu2
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.
N. Aydin and I. Siap, New quasi-cyclic codes over F5, Appl. Math. Lett., 15 (7), pp. 833–836, 2002.
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.
N. Bennenni, K. Guenda and S. Mesnager, DNA cyclic codes over rings, Adv. in Math. of Comm., 11 (1), pp. 83–98, 2017.
D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18(4), pp. 379–389, 2007.
R. Daskalov, P. Hristov, New binary one-generator quasi-cyclic codes, IEEE Trans. Inf. Theory, 49 (11), pp 3001–3005, 2003.
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.
Database of Z4 Codes. [online] Z4Codes. info (Accessed March, 2018).
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.
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.
J. Gao., Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform., 31 (3–4), pp. 337–342, 2013.
I. Siap and N. Kulhan, The Structure of Generalized Quasi Cyclic Codes, Appl. Math. E-Notes, vol. 5, pp. 24–30, 2005.
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.
J. Gao, F. Ma and F. Fu, Skew constacyclic codes over the ring Fq + vFq; Appl.Comput. Math., 6 (3), pp. 286–295, 2017 .
M. Grassl, Code Tables: Bounds on the parameters of codes, online, http://www.codetables.de/
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.
S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic over finite chain rings, Adv. Math.Commun., 6 (1), pp. 39–63, 2012.
P. Li, W. Dai and X. Kai, On Z2Z2[u]
Magma computer algebra system, online, http://magma.maths.usyd.edu.au/
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.
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.
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.
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.
T. Abualrub, I. Siap and I. Aydogdu, Z2(Z2 + uZ2)-Linear cyclic codes, Proceedings of the IMECS 2014, (2), Hong Kong, 2014.
T. Abualrub, I. Siap, and N. Aydin, Z2Z4
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.
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.
I. Aydogdu, T. Abualrub and I. Siap, Z2Z2[u]
N. Aydin and T. Asamov, A Database of Z4 Codes, Journal of Combinatorics, Information & System Sciences, 34 (1-4), pp. 1–12, 2009.
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.
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.
N. Aydin, Y. Cengellenmis and A. Dertli, On some constacyclic codes over Z4[u]=hu2
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.
N. Aydin and I. Siap, New quasi-cyclic codes over F5, Appl. Math. Lett., 15 (7), pp. 833–836, 2002.
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.
N. Bennenni, K. Guenda and S. Mesnager, DNA cyclic codes over rings, Adv. in Math. of Comm., 11 (1), pp. 83–98, 2017.
D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18(4), pp. 379–389, 2007.
R. Daskalov, P. Hristov, New binary one-generator quasi-cyclic codes, IEEE Trans. Inf. Theory, 49 (11), pp 3001–3005, 2003.
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.
Database of Z4 Codes. [online] Z4Codes. info (Accessed March, 2018).
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.
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.
J. Gao., Skew cyclic codes over Fp + vFp, J. Appl. Math. Inform., 31 (3–4), pp. 337–342, 2013.
I. Siap and N. Kulhan, The Structure of Generalized Quasi Cyclic Codes, Appl. Math. E-Notes, vol. 5, pp. 24–30, 2005.
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.
J. Gao, F. Ma and F. Fu, Skew constacyclic codes over the ring Fq + vFq; Appl.Comput. Math., 6 (3), pp. 286–295, 2017 .
M. Grassl, Code Tables: Bounds on the parameters of codes, online, http://www.codetables.de/
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.
S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic over finite chain rings, Adv. Math.Commun., 6 (1), pp. 39–63, 2012.
P. Li, W. Dai and X. Kai, On Z2Z2[u]
Magma computer algebra system, online, http://magma.maths.usyd.edu.au/
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.
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.
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.
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.
Published
2020-01-15
