1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$

Yıl 2022, Cilt: 9 Sayı: 1, 57 - 70, 15.01.2022

Arazgol Ghajari Kazem Khashyarmanesh Zohreh Rajabi

https://doi.org/10.13069/jacodesmath.1056624

Öz

In this paper, we obtain generating set of polynomials of two-dimensional cyclic codes over the ring $R=\mathbb{Z}_4[u]/\langle u^2-1\rangle$, where $u^2=1$. Moreover, we find generator polynomials for two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$ and specify a lower bound on minimum distance of free 1-generator two-dimensional quasi-cyclic codes and two-dimensional generalized quasi-cyclic codes over $R$.

Anahtar Kelimeler

Two-dimensional cyclic codes, Two-dimensional quasi-cyclic codes, Two-dimensional generalized quasi-cyclic codes

Kaynakça

• [1] Y. Cao, Structural properties and enumeration of 1-generator generalized quasi-cyclic codes, Des. Codes Cryptogr. 60(1) (2011) 67-79.
• [2] Y. Cao, J. Gao, Constructing quasi-cyclic codes from linear algebra theory, Des. Codes Cryptogr. 67(1) (2013) 59-75.
• [3] M. Esmaeili, S. Yari, Generalized quasi-cyclic codes: structural properties and code construction, Appl. Algebra Engrg. Comm. Comput 20(2) (2009) 159-173.
• [4] Y. Gao, J. Gao, T. Wu, F. W. Fu, 1-generator quasi-cyclic and generalized quasi-cyclic codes over the ring $\frac{\Bbb Z_4[u]}{\langle u^2-1\rangle}$, Appl. Algebra Engrg. Comm. Comput. 28(6) (2017) 457-467.
• [5] C. Guneri, F. $\ddot{\textmd{O}}$zbudak, B. $\ddot{\textmd{O}}$zkaya, E. Saçıkara, Z. Sepasdar, P. Sol$\acute{\textmd{e}}$, Structure and performance of generalized quasi-cyclic codes, Finite Fields Appl. 47 (2017) 183-202.
• [6] T. Ikai, H. Kosako, Y. Kojima, Two-dimensional cyclic codes, Electron. Comm. Japan 57(4) (1974/75) 27-35.
• [7] R. M. Lalasoa, R. Andriamifidisoa, T. J. Rabeherimanana, Basis of a multicyclic code as an ideal in $\Bbb F_q[X_1,\dots,X_s]/\langle X_1^{\rho_1}-1,\dots,X_s^{\rho_s}-1\rangle$, J. Algebra Relat. Topics 6(2) (2018) 63-78. [8] S. Ling, Ch. Xing, Coding theory: A first course, Cambridge University Press, New York (2004).
• [9] S. Ling, P. Sole, On the algebraic structure of quasi-cyclic codes. I. finite fields, IEEE Trans. Inform. Theory 47(7) (2001) 2751-2760.
• [10] M. Ozen, F. Z. Uzekmek, N. Aydin, N.T. $\ddot{\textmd{O}}$zzaim, Cyclic and some constacyclic codes over the ring $\frac{\Z_4[u]}{\langle u^2-1\rangle}$, Finite Fields Appl. 38 (2016) 27-39.
• [11] Z. Sepasdar, Generator matrix for two-dimensional cyclic codes of arbitrary length, arXiv:1704.08070v1 [math.AC] 26 Apr (2017).
• [12] Z. Sepasdar, Some notes on the characterization of two dimensional skew cyclic codes, J. Algebra Relat. Topics 4(2) (2016) 1-8.
• [13] Z. Sepasdar, K. Khashyarmanesh, Characterizations of some two-dimensional cyclic codes correspond to the ideals of $\Bbb{F}[x,y]/\langle x^s-1,y^{2^k}-1\rangle$, Finite Fields Appl. 41 (2016) 97-112.
• [14] I. Siap, T. Abualrub, B. Yildiz, One generator quasi-cyclic codes over $\F_2+u\F_2$, J. Frankl. Inst. 349(1) (2012) 284-292.