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

Previous Article
How to obtain division algebras used for fast-decodable space-time block codes
AMC Home
This Issue
Next Article
Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3)

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

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

Fatmanur Gursoy ^1, , Irfan Siap ^1, and Bahattin Yildiz ^2,

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

Abstract
Related Papers

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.

Keywords: codes over rings., Skew polynomial rings, skew cyclic codes.

Mathematics Subject Classification: Primary: 94B15, 94B05; Secondary: 94B9.

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), 2080-2090. doi: 10.1109/TIT.2010.2044062.
[2]	T. Abualrub and P. Seneviratne, Skew codes over rings, in Proc. IMECS, Hong Kong, 2010.
[3]	D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.
[4]	D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.
[5]	D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.
[6]	J. Gao, Skew cyclic codes over $\mathbb F_p+v\mathbb F_p$, J. Appl. Math. Inform., 31 (2013), 337-342. doi: 10.14317/jami.2013.337.
[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), 301-319. doi: 10.1109/18.312154.
[8]	S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 29-63. doi: 10.3934/amc.2012.6.39.
[9]	B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974.
[10]	I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inform. Coding Theory, 2 (2011), 10-20. doi: 10.1504/IJICOT.2011.044674.
[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), 1429-1432.
[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), 1680-1684. doi: 10.1109/TIT.2010.2040896.

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), 2080-2090. doi: 10.1109/TIT.2010.2044062.
[2]	T. Abualrub and P. Seneviratne, Skew codes over rings, in Proc. IMECS, Hong Kong, 2010.
[3]	D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.
[4]	D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.
[5]	D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.
[6]	J. Gao, Skew cyclic codes over $\mathbb F_p+v\mathbb F_p$, J. Appl. Math. Inform., 31 (2013), 337-342. doi: 10.14317/jami.2013.337.
[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), 301-319. doi: 10.1109/18.312154.
[8]	S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 29-63. doi: 10.3934/amc.2012.6.39.
[9]	B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc., New York, 1974.
[10]	I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inform. Coding Theory, 2 (2011), 10-20. doi: 10.1504/IJICOT.2011.044674.
[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), 1429-1432.
[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), 1680-1684. doi: 10.1109/TIT.2010.2040896.

[1]	Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[2]	Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[3]	Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[4]	Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005
[5]	Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[6]	Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[7]	Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[8]	Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[9]	Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022002
[10]	Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[11]	Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
[12]	Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[13]	Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[14]	Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[15]	Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043
[16]	Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41
[17]	Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012
[18]	Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022028
[19]	Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[20]	Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Fatmanur Gursoy Irfan Siap Bahattin Yildiz
on Google Scholar
Fatmanur Gursoy Irfan Siap Bahattin Yildiz

[Back to Top]