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

Year 2022, , 57 - 70, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056624

Abstract

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 $R$.

References

  • [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.
Year 2022, , 57 - 70, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056624

Abstract

References

  • [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.
There are 13 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Arazgol Ghajari This is me 0000-0002-3675-2832

Kazem Khashyarmanesh This is me 0000-0003-3314-7298

Zohreh Rajabi This is me 0000-0001-7857-2672

Publication Date January 15, 2022
Published in Issue Year 2022

Cite

APA Ghajari, A., Khashyarmanesh, K., & Rajabi, Z. (n.d.). 1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$. Journal of Algebra Combinatorics Discrete Structures and Applications, 9(1), 57-70. https://doi.org/10.13069/jacodesmath.1056624
AMA Ghajari A, Khashyarmanesh K, Rajabi Z. 1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):57-70. doi:10.13069/jacodesmath.1056624
Chicago Ghajari, Arazgol, Kazem Khashyarmanesh, and Zohreh Rajabi. “1-Generator Two-Dimensional Quasi-Cyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, no. 1 n.d.: 57-70. https://doi.org/10.13069/jacodesmath.1056624.
EndNote Ghajari A, Khashyarmanesh K, Rajabi Z 1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$. Journal of Algebra Combinatorics Discrete Structures and Applications 9 1 57–70.
IEEE A. Ghajari, K. Khashyarmanesh, and Z. Rajabi, “1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 57–70, doi: 10.13069/jacodesmath.1056624.
ISNAD Ghajari, Arazgol et al. “1-Generator Two-Dimensional Quasi-Cyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/1 (n.d.), 57-70. https://doi.org/10.13069/jacodesmath.1056624.
JAMA Ghajari A, Khashyarmanesh K, Rajabi Z. 1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$. Journal of Algebra Combinatorics Discrete Structures and Applications.;9:57–70.
MLA Ghajari, Arazgol et al. “1-Generator Two-Dimensional Quasi-Cyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 57-70, doi:10.13069/jacodesmath.1056624.
Vancouver Ghajari A, Khashyarmanesh K, Rajabi Z. 1-generator two-dimensional quasi-cyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):57-70.