Skew cyclic codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$
DOI:
https://doi.org/10.13069/jacodesmath.90080Keywords:
Linear codes, Skew cyclic codes, Gray map, Generator polynomialAbstract
In this paper, we study skew cyclic codes over the ring $R=\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, where $u^{2}=u,v^{2}=v,uv=vu$, $q=p^{m}$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $R$ through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length $n$ over $R$.
Received: 25 March 2015 | Accepted: 11 July 2015
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References
T. Abualrub and P. Seneviratne, Skew codes over rings, in Proceedings of the International MultiConference of Engineers and Computer Scientists (IMECS 2010), Hong Kong, Vol. II (2010) 955-958.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics 13, Springer-Verlag, New York (1992).
D. Boucher, W. Geiselmann, and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18(4) (2007) 379-389.
D. Boucher and F. Ulmer, Coding with skew polynomial ring, J. Symbolic Comput. 44(12) (2009) 1644-1656.
D. Boucher, P. Solé, and F. Ulmer, Skew constacyclic codes over Galois ring, Adv. Math. Commun. 2(3) (2008) 273-292.
J. Gao, Skew cyclic codes over $\mathbb{F}_p + v\mathbb{F}_p$, J. Appl. Math. & Informatics 31(3-4) (2013) 337-342.
J. Gao, L. Shen, and F.-W. Fu, Skew generalized quasi-cyclic codes over finite fields, Cryptogr. Commun. 8(3) (2016) 429-445.
F. Gürsoy, I. Siap, and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_q + v\mathbb{F}_q$, Adv. Math. Commun. 8(3) (2014) 313-322.
F. Hernando and D. Ruano, Sixteen new linear codes with Plotkin sum, arXiv:0804.3507 [math.IT] (2008).
S. Jitman, S. Ling, and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun. 6(1) (2012) 29-63.
A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319.
I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2(1) (2011) 10-20.
Y. T. Zhang, Research on constacyclic codes over some classes of finite non-chain rings, Master's thesis, Hefei University of Technology (2013).
