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

Abstract

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 R k−1,m and cyclic codes of length n over the ring R k,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.

Cyclic codes over R3,m

“…One of the non-chain rings that has been considered recently in coding theory is the ring R k,m = F 2 m [u1,...,u k ] codes of arbitrary length n over the ring R 3,m = F 2 m + uF 2 m + vF 2 m + wF 2 m + uvF 2 m + uwF 2 m + vwF 2 m + uvwF 2 m , defined as a characteristic 2 ring subject to the restrictions u 2 = v 2 = w 2 = 0, uv = vu, uw = wu, vw = wv, and generalize the results of [14] to R 3,m . It seems that it is possible to generalize these results to R k,m , where k is arbitrary, which is our motivation for doing this work.…”

“…We also determined all ideals of the ring R 3,m,2 . Note that we can determine all ideals of the ring R 4,m by [14,Corollary 2]. In fact, the Sec.…”

“…). If f 1 = x − 1 then we consider the following cases: Similarly, conditions (1), (3), (4), and (9) imply f 4 | k 2 , conditions (1), (3), (4), (5), (9), (10), and (13) imply f 5 | k 3 , conditions (1), (3), (4), (5), (6), (9), (10), (13) and (16) imply f 6 | k 4 , conditions (1), (3), (4), (5), (6), (7), (9), (10), (12), (13), (14), (16), (17), and (18) imply f 7 | k 5 , conditions (1), (3), . .…”

“…Codes over finite rings have been studied extensively since the rediscovery of good nonlinear codes from extended linear cyclic codes over Z 4 , [11]. For example, codes over finite chain rings and finite non-chain rings are being widely studied, [1][2][3][4][5][6][7][8][14][15][16].…”

“…3, we try to determine all distinct cyclic codes of length 2 over the ring R 3,m by the previous section. Also note that it is easy to determine all ideals of the ring R 4,m by a one-to-one correspondence between cyclic codes of length 2 over the ring R 3,m and cyclic codes of length 1 over the ring R 4,m , see [14,Corollary 2], note that the determination of all ideals of the ring R k,m was introduced as a challenging open problem in [10].…”