Some results on cyclic codes over the ring R2,m

Abstract

Let Rk,m be the ring F2m[u1,u2, . . . ,uk/〈u2i,uiUj - uiUj〉. 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-i,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. © Tübi̇tak.