Construction of various codes over finite rings

Keywords: Cyclic code, constacyclic code, quasi cyclic code, skew cyclic code, skew constacyclic code, DNA code, reversible DNA code, addtitive code This study consists of six sections. In the first section, algebraic definitions and theorems are included under various titles. In the second section, by examining the structure of the ring Z4+uZ4+u^2Z4+...+u^(k-1)Z4, with the help of the Chinese Remainder Theorem, the structure of cyclic and constacyclic codes with odd length over this ring is investigated. Generator polynomials are constructed and the idempotent generator of the code is shown to be unique. Important results have been obtained by examining the Z4-images of the cyclic and constacyclic codes over the ring. At the end, many new and optimal linear codes are obtained. Some of these codes are presented in tables. In the third section, all non-trivial automorphisms over the ring Z4+uZ4+u^2Z4 are determined. With the help of the Chinese Remainder Theorem, generator polynomials of skew cyclic and skew constacyclic codes of odd length are constructed and their characteristic structures are analyzed. Many new and optimal linear codes have been obtained by examining Z4-images of skew cyclic codes and skew constacyclic codes for all unit elements over the ring. The codes that found are presented with tables. In the fourth section, the existence of reversible DNA codes over the ring Z4+uZ4+u^2Z4 is focused and a new polynomial is defined to find these codes. The relationship is established between DNA 2-mers and the Gray map in the previous sections. Then, a new construction method is presented to obtain the reversible DNA code over this ring. In order to make this method more understandable, examples are given. In the fifth section, the structural properties of the ring Z4Z4[u] are explored and the generator polynomials of the cyclic and constacyclic codes are identified. By defining an automorphism, the structure of skew cyclic and skew constacyclic codes are investigated. New results have been obtained via analyzing Z4-images of cyclic, constacyclic, skew cyclic and skew constacyclic over this ring. In the sixth section, the results are given.