Following the very recent studies on ℤ2ℤ4-additive codes, ℤ2ℤ2[u]-linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R-module Z2αRβwhere R = ℤ2+uℤ2= {0, 1, u, u + 1} is the ring with four elements and u2= 0. We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in Z2αRβ.

In this paper, we study Z2Z4-additive cyclic codes. These codes are identified as Z4[x]-submodules of the ring Rr,s=Z2[x]/〈xr-1〉×Z4[x]/〈xs-1〉. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a Z4[x]-submodule of the ring Rr,sis determined. We show that the duals of Z2Z4-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the Z2Z4-additive cyclic codes.

In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ringF[x;¿]. After a brief description of the skew polynomial ringF[x;¿], it is shown that skew QC codes are left submodules of the ringRsl=(F[x;¿]/(xs-1) )l. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ringF[x;¿] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.

In this paper, we study cyclic codes of length n= 2/sup e/ over the ring R/sub 4/= /spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, we derive a mass formula of these codes for a given length n. We also give an example in which we study codes of length 8.

In this correspondence, we study cyclic codes of length n=2/sup e/ over the ring R/sub 4/=/spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, we derive a closed-form expression for the number of these codes for a given length n. We also study the rank of these codes and derive an expression for that. Furthermore, we give an example in which we study all cyclic codes of length 8. We also study all self-dual codes of length 8 and 16 and classify them according to their type.

Results are presented on the generators of ideals in the ring /spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, each ideal (cyclic code) has a unique distinguished set of generators that characterizes any cyclic code. Some results about dual codes are also included.