Dergiler / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 6
Some results on cyclic codes over the ring R2,m
- Sayfa
- 1061–1074
- DOI
- —
Özet
Let Rk,m be the ring F2m[u1,u2,...,uk]/\gen ui2, uiuj-ujui. In this paper, cyclic codes of arbitrary length n over the ring R2,m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m-basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n, n odd, over the ring Rk-1,m and cyclic codes of length n over the ring Rk,m. By determining the complete structure of cyclic codes of length 2 over R2,m, a mass formula for the number of these codes is given. Using this and the mentioned correspondence, the number of ideals of the rings R2,m and R3,m is determined. As a corollary, the number of cyclic codes of odd length n over the rings R2,m and R3,m is obtained.
Abstract
Let Rk,m be the ring F2m[u1,u2,...,uk]/\gen ui2, uiuj-ujui. In this paper, cyclic codes of arbitrary length n over the ring R2,m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m-basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n, n odd, over the ring Rk-1,m and cyclic codes of length n over the ring Rk,m. By determining the complete structure of cyclic codes of length 2 over R2,m, a mass formula for the number of these codes is given. Using this and the mentioned correspondence, the number of ideals of the rings R2,m and R3,m is determined. As a corollary, the number of cyclic codes of odd length n over the rings R2,m and R3,m is obtained.