[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2013.50.5.1513

CYCLIC CODES OVER SOME SPECIAL RINGS

Flaut, Cristina (Faculty of Mathematics and Computer Science Ovidius University)

Publication Information

Bulletin of the Korean Mathematical Society / v.50, no.5, 2013 , pp. 1513-1521 More about this Journal

Abstract

In this paper we will study cyclic codes over some special rings: $$\mathbb{F}_q[u$ $]$ $/(u^i)$$ , $$\mathbb{F}_q[u_1,{\ldots},u_i$ $]$ $/(u^2_1,u^2_2,{\ldots},u^2_i,u_1u_2-u_2u_1,{\ldots},u_ku_j-u_ju_k,{\ldots})$$ , and $$\mathbb{F}_q[u,v$ $]$ $/(u^i,v^j,uv-vu)$$ , where $\mathbb{F}_q$ is a field with $q$ elements $q=p^r$ for some prime number $p$ and $r{\in}\mathbb{N}-\{0\}$ .

Keywords

cyclic codes; codes over rings; Hamming distance;

Citations & Related Records

Reference

1	A. A. Nechaev and T. Honold, Fully weighted modules and representations of codes, (Russian) Problemy Peredachi Informatsii 35 (1999), no. 3, 18-39; translation in Prob-lems Inform.Transmission 35 (1999), no. 3, 205-223.
2	J.-F. Qian, L.-N. Zhang, and A.-X. Zhu, Cyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+{\cdot}{\cdot}{\cdot}+u^{k-1}{\mathbb{F}}_p$ , IEICE Trans. Fundamentals Vol. E88-A (2005), no. 3, 795-797. DOI
3	P. Sole and V. Sison, Bounds on the minimum homogeneous distance of the $p^r$ -ary image of linear block codes over the Galois ring GR( $p^r$ ,m), IEEE Trans. Inform. Theory 53 (2007), no. 6, 2270-2273. DOI ScienceOn
4	P. Sole and V. Sison, Quaternary convolutional codes from linear block codes over Galois rings, IEEE Trans. Inform. Theory 53 (2007), no. 6, 2267-2270. DOI ScienceOn
5	E. Spiegel, Codes over $Z_m$ revisited, Inform. and Control 37 (1978), no. 1, 100-104. DOI
6	B. Yildiz and S. Karadeniz, Cyclic codes over ${\mathbb{F}}_2+u{\mathbb{f}}_2+u{\mathbb{f}}_2+uu{\mathbb{f}}_2$ , Des. Codes Cryptogr. 58 (2011), no. 3, 221-234. DOI
7	T. Abualrub and I. Siap, On the construction of cyclic codes over the ring ${\mathbb{Z}}_2+u{\mathbb{Z}}_2$ , WSEAS Trans. Math. 5 (2006), no. 6, 750-755.
8	T. Abualrub and I. Siap, Cyclic codes over the rings ${\mathbb{Z}}_2+u{\mathbb{Z}}_2\;and\;{\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2$ , Des Codes Cryptogr. 42 (2007), no. 3, 273-287. DOI
9	M. M. Al-Ashker and M. Hamoudeh, Cyclic codes over ${\mathbb{Z}}_2+u{\mathbb{Z}}_2{\cdot}{\cdot}{\cdot}u^{k-1}{\mathbb{Z}}_2$ , Turk J. Math. 34 (2010), 1-13.
10	M. Bhaintwal and S. K. Wasan, On quasi-cyclic codes over Zq, Appl. Algebra Engrg. Comm. Comput. 20 (2009), no. 5-6, 459-480. DOI
11	I. F. Blake, Codes over certain rings, Inf. Control 20 (1972), 396-404. DOI ScienceOn
12	S. T. Dougherty, S. Karadeniz, and B. Yildiz, Cyclic codes over $R_k$ , Des. Codes Cryptogr. 63 (2012), no. 1, 113-126. DOI
13	S. T. Dougherty, H. Liu, and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011), 39-53.
14	D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, Berlin, New York, 1995.
15	M. Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), no. 1-3, 273-277. DOI ScienceOn
16	J. G. Milne, Etale cohomology, Princeton University Press, 1980.
17	A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, The $Z_4$ liniarity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301-319. DOI ScienceOn
18	T. W. Hungerford, Algebra, Springer Verlag, New York, 1974.
19	B. R. McDonald, Finite Rings with Identity, New York, Marcel Dekker Inc., 1974.