Cyclic codes over $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$

Abstract

In this paper, we study cyclic codes over the ring R = Z4 + uZ4 + u 2 Z4 , where u 3 = 0 . We investigate Galois extensions of this ring and the ideal structure of these extensions. The results are then used to obtain facts about cyclic codes over R . We also determine the general form of the generator of a cyclic code and find its minimal spanning sets. Finally, we obtain many new linear codes over Z4 by considering Gray images of cyclic codes over R .

Cyclic DNA codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$

“…Ozen et al in [15] have determined cyclic codes over the ring R = Z 4 + uZ 4 + u 2 Z 4 , where u 3 = 0. They obtained that the total number of cyclic codes over R is 13 r .…”

“…Similarly, we get p 3 (x) = q 3 (x). Now, we study the reverse constraint of one generator cyclic code of odd length n over R. For this, first we need the following result which states one generator cyclic codes of odd length n over the ring R. This result with its proof can be found in [15].…”

“…Abhay et al [10] have studied DNA cyclic codes over the ring Z 4 + uZ 4 with u 2 = 0. Cyclic codes over the ring R = Z 4 + uZ 4 + u 2 Z 4 with u 3 = 0 have been studied by Ozen et al in [15]. Cyclic code over skew polynomial ring have been constructed by [7].…”

“…[15] Let C be a cyclic code of odd length n over R.If C = h 1 (x) + ug 1 (x) + u 2 b 1 (x), uh 2 (x) + u 2 b 2 (x), u 2 h 3 (x)and h 1 (x) and h 3 (x) are equal, thenC = h 1 (x) + ug 1 (x) + u 2 b 1 (x) .Definition 3.6. Let R be a ring and p(x) ∈ R[x] is called regular polynomial if it is not a zero divisor in R[x].…”

Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

“…Kumar et al [16] concentrated the DNA construction and binary images of DNA codes over the ring Z 4 +vZ 4 when v 2 = v . Besides all these studies, if we talked about the various rings with 64 elements; Özen et al [19] investigated the structure of the ring Z 4 + uZ 4 + u 2 Z 4 where u 3 = 0 and Galois extensions of this ring. They also studied the ideal structure of these extensions and used their results to obtain for the cyclic codes over this ring.…”