M-adic residue codes over fq[v]/(v^2-v) and DNA codes

At a time when almost all of the information storage and transmission is through digital media, it is very important to detect and correct the errors in data transmission. Algebraic coding theory, which provides a mathematical solution to this subject, develops rapidly and highly efficient methods are built. The basic goal of this area is to obtain codes with optimal parameters. The need for error detection and correction is not only in digital media, but also in other areas. DNA is one of these areas. Detection and correction of errors on DNA is a problem that has become popular in recent times. Researchers are trying to solve this problem with the help of DNA codes. The relations that occur on DNAs are tried to be solved by using the correlation between DNAs and optimal codes over fields. In this thesis, the m-adic residue codes over the fields are defined on some non-chain rings by means of idempotent generators and optimal codes are obtained. In addition, these codes are associated with DNA codes and a simpler and more comprehensive method than the previously described DNA code generation methods has been constructed. In the first chapter, the relevant literature summary, the thesis purpose, the innovations brought by the thesis and the hypothesis will be given. In the second chapter, the field theory will be given. In the third chapter, basic informations about algebraic coding theory will be given. In the fourth chapter, the definition of the quadratic codes and some examples will be given. In the fifth chapter, definition of the qth power residue codes, which is a general form of quadratic codes, will be given, and the general structure of the idempotent generators on this ring will be determined. In the sixth chapter, the definition of the m-adic residue codes and some examples of the code will be given. In the seventh chapter, the m-adic residue codes over the ring Fq[v]/(v^2-v) will be given by idempotent generators, and the necessary conditions for the generator polynomials to be palindromic will be determined. In addition, some optimal codes over the ring with respect to Griesmer bound will be given in this chapter. In the eighth chapter, by means of the palindromic generator polynomials we obtained in the seventh chapter, reversable DNA codes over the ring F4^2k[v]/(v^2-v) will be built and an example will be given. In the last chapter, the results obtained will be summarized and some suggestions will be given.