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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]