Cyclic codes over noncommutative rings

Coding theory is a field of research with a focus on detection and correction of errors that can occur during the transmission of data or data storing. It is a multi-disciplinary field with a lot of connections to different areas of mathematics. One of the main tasks in coding theory is to encode messages with minimum cost and maximum error correction capability. Early studies in this area were concentrated on linear and cyclic codes over fields. Some of the important families of codes obtained in early stages were Hamming codes, BCH codes and Golay codes. Codes over rings had been considered by mathematicians from early seventies, but the breakthrough work was published in 1994 by Hammons et al. in which they showed that some important binary nonlinear codes such as Kerdock and Preparata codes can be obtained as Gray images of linear codes over Z_4. The emergence of this paper brought a new direction to researchers working in coding theory. Since then a lot of research has been directed towards codes over rings. In the last six years cyclic codes from noncommutative polynomial rings have been introduced and the algebraic structure of these codes has been examined. This new class of codes are named skew cyclic codes and they are important because of their algebraic structure. Since skew polynomial rings are not unique factorization rings there are many more generator polynomials leading to many more skew cyclic codes compared to ordinary cyclic codes of the same lengths. Therefore skew cyclic codes are advantageous to search for codes with possible good parameters. Another important aspect of studying codes over such structures is the fact that one obtains a representation of linear codes over fields with a richer algebraic structure. In this work, studies about skew cyclic codes in literature were examined and exemplified. Idempotent generators of skew cyclic codes are identified and it is shown that idempotent generator of a skew cyclic code may not be unique. The characteristic 2 ring F_4+vF_4 of size 16 is considered. The properties of this ring are studied and linear codes over this ring are introduced. Also a Gray map is defined over this ring and codes over F_4 obtained as Gray images. Skew-cyclic codes over this ring are considered for the first time in the literature. The algebraic properties of these codes are examined and some good codes are obtained through the images of these codes.