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In this paper, a new class of additive codes which is referred to as ℤ2ℤ2[u]-additive codes is introduced. This is a generalization towards another direction of recently introduced ℤ2ℤ4-additive codes [J. Borges, C. Fernández-Córdoba, J. Pujol, J. Rif´a, and M. Villanueva, ℤ2ℤ4-linear codes: Generator matrices and duality, Designs Codes Cryptogr. 54(2) (2010), pp. 167–179]. ℤ2ℤ4-additive codes have shown to provide a promising class of codes with their algebraic structure and applications such as steganography. The standard generator matrices are established and by introducing orthogonality the parity-check matrices are also obtained. A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, a Gray map that maps these codes to binary codes is defined and some examples of optimal codes which are the binary Gray images of ℤ2ℤ2[u]-additive codes are presented. [ABSTRACT FROM PUBLISHER]

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 ring F[x;θ]. After a brief description of the skew polynomial ring F[x;θ], it is shown that skew QC codes are left submodules of the ring Rsl = (F[x;θ]/(xs - 1))l. The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring F[x;θ] 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. [ABSTRACT FROM AUTHOR]

In this correspondence, we study cyclic codes of length n = 2e over the ring R4 = Z4 [x]/(xn - 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. [ABSTRACT FROM AUTHOR]

Results are presented on the generators of ideals in the ring Z[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. [ABSTRACT FROM AUTHOR]

Cyclic codes of odd length over Z4 have been studied by many authors. But what is the form of cylic codes of even length? The structure of cyclic codes of length n=2e, for any positive integer e is considered. We show that any cyclic code is an ideal in the ring Rn=Z4[x]/〈xn−1〉. We show that the ring Rn is a local ring but not a principal ideal ring. Also, we find the set of generators for cyclic codes. Examples of cyclic codes of such length are given. [Copyright &y& Elsevier]