Some results on cyclic codes over the ring $R_{2,m}$

Abstract

Let Rk;m be the ring F2m(u1;u2; � � � ;uk)= ⟨ u 2;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.