# CYCLIC CODES OVER SOME SPECIAL RINGS

• Flaut, Cristina
• Published : 2013.09.30
• 48 9

#### 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

#### References

1. 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.
2. 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. https://doi.org/10.1007/s10623-006-9034-5
3. 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.
4. M. Bhaintwal and S. K. Wasan, On quasi-cyclic codes over Zq, Appl. Algebra Engrg. Comm. Comput. 20 (2009), no. 5-6, 459-480. https://doi.org/10.1007/s00200-009-0110-8
5. I. F. Blake, Codes over certain rings, Inf. Control 20 (1972), 396-404. https://doi.org/10.1016/S0019-9958(72)90223-9
6. S. T. Dougherty, S. Karadeniz, and B. Yildiz, Cyclic codes over $R_k$, Des. Codes Cryptogr. 63 (2012), no. 1, 113-126. https://doi.org/10.1007/s10623-011-9539-4
7. S. T. Dougherty, H. Liu, and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011), 39-53.
8. D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, Berlin, New York, 1995.
9. M. Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), no. 1-3, 273-277. https://doi.org/10.1016/S0012-365X(97)00006-X
10. 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. https://doi.org/10.1109/18.312154
11. T. W. Hungerford, Algebra, Springer Verlag, New York, 1974.
12. B. R. McDonald, Finite Rings with Identity, New York, Marcel Dekker Inc., 1974.
13. J. G. Milne, Etale cohomology, Princeton University Press, 1980.
14. 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.
15. 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. https://doi.org/10.1093/ietfec/e88-a.3.795
16. 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. https://doi.org/10.1109/TIT.2007.896891
17. 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. https://doi.org/10.1109/TIT.2007.896884
18. E. Spiegel, Codes over $Z_m$ revisited, Inform. and Control 37 (1978), no. 1, 100-104. https://doi.org/10.1016/S0019-9958(78)90461-8
19. 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. https://doi.org/10.1007/s10623-010-9399-3