A note on two-dimensional cyclic and constacyclic codes
DOI:
https://doi.org/10.13069/jacodesmath.v9i3.185Keywords:
Cyclic codes, Two-dimensional cyclic codes, Constacyclic codes, Dual codes, Generator matrixAbstract
During the study of the two-dimensional cyclic (TDC) codes of length $n=ls$ over a finite field $\mathbb{F}_q$ where $s=2^k$, Sepasdar and Khashyarmanesh (2016, [11]) arose a problem that the technique used by them to characterize TDC codes of length $n=ls$ does not work for TDC codes of length $3l$. It naturally motivates us to study the TDC codes of other lengths together with $3l$. Further, $(\lambda_1,\lambda_2)$-constacyclic codes are the generalization of constacyclic codes. Thus, we study two-dimensional cyclic codes of length $3l$ and $(\lambda_1,\lambda_2)$-constacyclic codes of length $2l$, respectively over finite fields. Here, the generating set of polynomials for these two-dimensional codes and their duals are obtained. Finally, with the help of our derived results, we have constructed many MDS codes corresponding to the two-dimensional codes.
Received: 11 January 2021 | Accepted: 31 March 2022Downloads
References
W. Bosma and J. Cannon, Handbook of Magma Functions, University of Sydney, Sydney, 1995.
I. F. Blake, Codes over certain rings, Inform. and Control 20(4) (1972) 396–404.
I. F. Blake, Codes over integer residue rings, Inform. and Control 29(4) (1975) 295–300.
H. Imai, A theory of two-dimensional cyclic codes, Inform. and Control 34(1) (1977) 1–21.
T. Ikai, H. Kosako, and Y. Kojima, Two-dimensional cyclic codes, Electron. Comm. Japan 57(4) (1974/75) 27–35.
S. Ling and C. Xing, Coding Theory: A First Course, Cambridge University Press, Cambridge, 2004.
S. Patel and O. Prakash, Repeated-root bidimensional $(\mu,\nu)$-constacyclic codes of length $4p^t \cdot 2^r$, Int. J. Inf. Coding Theory 5(3/4) (2020) 266–289.
S. Patel, O. Prakash, and H. Islam, Cyclic codes over $M_4(\mathbb{F}_2 + u\mathbb{F}_2)$, Cryptogr. Commun. 14(5) (2022) 1021–1034.
E. Prange, Cyclic error-correcting codes in two symbols, Tech. Rep. AFCRC-TN-57-103, Air Force Cambridge Research Center, Cambridge, MA, 1957.
Z. Rajabi and K. Khashyarmanesh, Repeated-root two-dimensional constacyclic codes of length $2p^s \cdot 2^k$, Finite Fields Appl. 50 (2018) 122–137.
Z. Sepasdar and K. Khashyarmanesh, Characterizations of some two-dimensional cyclic codes correspond to the ideals of $\mathbb{F}[x,y]/\langle x^s - 1, y^{2^k} - 1 \rangle$, Finite Fields Appl. 41 (2016) 97–112.
A. Sharma and M. Bhaintwal, A class of 2D skew-cyclic codes over $\mathbb{F}_q + u\mathbb{F}_q$, Appl. Algebra Engrg. Comm. Comput. 30(6) (2019) 471–490.
X. Li and H. Li, 2-D skew-cyclic codes over $\mathbb{F}_q[x,y;\rho,\theta]$, Finite Fields Appl. 25 (2014) 49–63.
B. Yildiz and N. Aydin, On cyclic codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory 2(4) (2014) 226–237.
X. Zheng and B. Kong, Cyclic codes and $\lambda_1 + \lambda_2 u + \lambda_3 v + \lambda_4 uv$-constacyclic codes over $\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p$, Appl. Math. Comput. 306 (2017) 86–91.
