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In this paper, we use a novel approach to describe generator polynomials of quasi-cyclic (QC) and generalized QC (GQC) codes over finite fields. Our study of QC- and GQC-codes will be general and not only restricted to one-generator codes. We prove that generator polynomials of QC-codes and GQC-codes are unique. Further, we use our results to obtain an expression for the dimensions of QC-codes and GQC-codes. As an application of our construction of these codes, we obtain many optimal linear codes over finite fields GF (2) , GF (3) , GF (4) and GF (5). [ABSTRACT FROM AUTHOR]

ℤ 2 ℤ 2 s -additive codes for any integer s > 1 are considered as codes over mixed alphabets. They are a generalization of binary linear codes and linear codes over ℤ 2 s . In this paper, we are interested in studying ℤ 2 ℤ 2 s -additive cyclic codes. We will give the generator polynomials of these codes. We will also give the minimal spanning sets for these codes. We will define separable ℤ 2 ℤ 2 s -additive codes and provide conditions on the generator polynomials for a ℤ 2 ℤ 2 s -additive cyclic code to be separable. Finally, we present some examples of optimal parameter binary codes obtained as images of ℤ 2 ℤ 8 -additive cyclic codes. [ABSTRACT FROM AUTHOR]

Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field , obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class. [ABSTRACT FROM AUTHOR]

This article studies skew generalized quasi-cyclic codes (SGQC-codes) over finite fields for any length n. We derive generator polynomials and cardinality of SGQC codes. Moreover, we show that the dual of any length SGQC-code is also an SGQC-code. Our search results lead to the construction of fifteen new 2-generator SGQC codes over the finite field F 4 with minimum distances exceeding the minimum distances of the previously best known F 4 -linear codes with comparable parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study \BBZ2\BBZ4-additive cyclic codes. These codes are identified as \BBZ4[x]-submodules of the ring Rr,s=\BBZ2[x]/\langle x^{r}-1\rangle\times\BBZ4\left[x\right]/\langle x^{s}-1\rangle. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a \BBZ4[x]-submodule of the ring Rr,s is determined. We show that the duals of \BBZ2\BBZ4-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 \BBZ2\BBZ4-additive cyclic codes. [ABSTRACT FROM AUTHOR]

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]

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]