Coding Theory over Rings

Coding theory is a very relevant and researched science in the modern digital world. Various classes of codes for encoding information are used in all areas of data transmission: from reading information from CDs to receiving and transmitting messages from probes within the solar system. In my work, I decided to consider in particular detail such classes of codes as: - dual, - self-dual, - double even self-dual. Despite the fact that these codes are the most important classes and have been known for a long time, they remain not fully explored, and in the scientific segment of articles (including Internet articles), especially Russian, there is little information about these classes of codes. I wanted, first of all, to present all the materials found in an accessible way and explain with simple examples what these codes are and what their features are. In this paper, we investigate a new algebraic approach for the classification of self-dual codes based on group theory. More precisely, we study the action of an orthogonal group preserving the Hamming metric on a set of codes. The work for the most part describes coding over F2 fields and building codes over fields. But I suppose that similar methods of group theory can be applied to classify linear codes for rings, which is why the work is called "Coding over rings". This will require a detailed study of the action of orthogonal groups of finite quadratic forms on finite discriminant groups (i.e., discriminant forms).