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.

One of the most important problems of coding theory is to construct codes with the best possible minimum distance. We further generalize the method first introduced by Gulliver and Harada (see Des., Codes Cryptogr, vol. 22, no. 1, p.89-96, 2001) and later generalized by the present authors, and obtain new linear codes which improve the best known minimum-distance bounds of certain linear codes. We have found eight new linear codes over F/sub 5/ with improved minimum distances. We introduce a generalized version of a Gray map, then we give definitions of quasi- and nearly quasi-cyclic codes. We conclude by giving the parameters of new linear codes with their generator matrices.

One of the most important problems of coding theory is to construct codes with best possible minimum distances. In this article, using the structure of quasi-cyclic (QC) codes with a BCH-type bound given by Lally and Fitzpatrick (see WCC '99, Workshop on Coding and Cryptography, Paris, France, 1999), we have found 20 new ternary quasi-cyclic codes with improved minimum distances.