August  2014, 8(3): 313-322. doi: 10.3934/amc.2014.8.313

Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$

1. 

Department of Mathematics, Yildiz Technical University, 34210, Istanbul, Turkey, Turkey

2. 

Department of Mathematics, Fatih University, 34500, Istanbul

Received  May 2013 Revised  July 2013 Published  August 2014

In this paper skew cyclic codes over the the family of rings $\mathbb{F}_q+v\mathbb{F}_q$ with $v^2=v$ are studied for the first time in its generality. Structural properties of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$ are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyclic codes over $\mathbb{F}_q$ and $\mathbb{F}_q+v\mathbb{F}_q$ have been considered for the first time in literature. Moreover, a BCH type bound is presented for the parameters of these codes.
Citation: Fatmanur Gursoy, Irfan Siap, Bahattin Yildiz. Construction of skew cyclic codes over $\mathbb F_q+v\mathbb F_q$. Advances in Mathematics of Communications, 2014, 8 (3) : 313-322. doi: 10.3934/amc.2014.8.313
References:
[1]

T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap, On the construction of skew quasi-cyclic codes,, IEEE Trans. Inform. Theory, 56 (2010), 2080.  doi: 10.1109/TIT.2010.2044062.  Google Scholar

[2]

T. Abualrub and P. Seneviratne, Skew codes over rings,, in Proc. IMECS, (2010).   Google Scholar

[3]

D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes,, Appl. Algebra Eng. Comm., 18 (2007), 379.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[4]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273.  doi: 10.3934/amc.2008.2.273.  Google Scholar

[5]

D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symb. Comput., 44 (2009), 1644.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[6]

J. Gao, Skew cyclic codes over $\mathbb F_p+v\mathbb F_p$,, J. Appl. Math. Inform., 31 (2013), 337.  doi: 10.14317/jami.2013.337.  Google Scholar

[7]

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 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[8]

S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings,, Adv. Math. Commun., 6 (2012), 29.  doi: 10.3934/amc.2012.6.39.  Google Scholar

[9]

B. R. McDonald, Finite Rings with Identity,, Marcel Dekker Inc., (1974).   Google Scholar

[10]

I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length,, Int. J. Inform. Coding Theory, 2 (2011), 10.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar

[11]

X. Q. Xu and S. X. Zhu, Skew cyclic codes over the ring $\mathbb F_4+v\mathbb F_4$,, J. Hefei Univ. Technol. Nat. Sci., 34 (2011), 1429.   Google Scholar

[12]

S. Zhu, Y. Wang and M. Shi, Some results on cyclic codes over $\mathbb F_2+v\mathbb F_2$,, IEEE Trans. Inform. Theory, 56 (2010), 1680.  doi: 10.1109/TIT.2010.2040896.  Google Scholar

show all references

References:
[1]

T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap, On the construction of skew quasi-cyclic codes,, IEEE Trans. Inform. Theory, 56 (2010), 2080.  doi: 10.1109/TIT.2010.2044062.  Google Scholar

[2]

T. Abualrub and P. Seneviratne, Skew codes over rings,, in Proc. IMECS, (2010).   Google Scholar

[3]

D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes,, Appl. Algebra Eng. Comm., 18 (2007), 379.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[4]

D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273.  doi: 10.3934/amc.2008.2.273.  Google Scholar

[5]

D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symb. Comput., 44 (2009), 1644.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[6]

J. Gao, Skew cyclic codes over $\mathbb F_p+v\mathbb F_p$,, J. Appl. Math. Inform., 31 (2013), 337.  doi: 10.14317/jami.2013.337.  Google Scholar

[7]

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 (1994), 301.  doi: 10.1109/18.312154.  Google Scholar

[8]

S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings,, Adv. Math. Commun., 6 (2012), 29.  doi: 10.3934/amc.2012.6.39.  Google Scholar

[9]

B. R. McDonald, Finite Rings with Identity,, Marcel Dekker Inc., (1974).   Google Scholar

[10]

I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length,, Int. J. Inform. Coding Theory, 2 (2011), 10.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar

[11]

X. Q. Xu and S. X. Zhu, Skew cyclic codes over the ring $\mathbb F_4+v\mathbb F_4$,, J. Hefei Univ. Technol. Nat. Sci., 34 (2011), 1429.   Google Scholar

[12]

S. Zhu, Y. Wang and M. Shi, Some results on cyclic codes over $\mathbb F_2+v\mathbb F_2$,, IEEE Trans. Inform. Theory, 56 (2010), 1680.  doi: 10.1109/TIT.2010.2040896.  Google Scholar

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