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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]