Additive codes on some special modules

Today, which is the age of technology; communication devices are very important. Every day, we come in contact with mobile phones, internet, television etc. and use other various modern communication systems. Using these media, we instantly come in contact with people in different cities and countries and also we are instantly informed about events that occur around the world through television and internet. By using e-mail and other social media we can send messages to our friends and family we even can chat with video call. Although we cannot imagine a world without these means of communications, a large part of these important technologies do not have a very ancient history. Together with the phone came into our lives in 1870, the communication technology has advanced day by day and reached to today's status. Dissemination of the communication devices brought about a need for a securer and quality communication and a branch of mathematics called information and coding theory began to seek a solution for this need. The most important initial development in information theory and coding theory was in 1948, when Shannon published a remarkable paper [1]. In this paper, Shannon identified a general mechanism for a communication system and set a base for information and coding theory. In 1990's the study of coding theory began to transfer onto the rings. In particular, studies about the ring and finding the binary images of these codes by defining a special Gray map was an initial point for studies on finite rings. In 2010, additive codes were introduced and the structures of these codes were determined. These families of codes are interesting and have become an open research area of coding theory since they are related with both binary and quaternary codes. This thesis generalizes the results related to additive codes which have been introduced so far. In the first chapter of the thesis, which is the introduction part, we mention about studies on additive codes covered in the literature and we state the goal of this thesis. The second chapter covers the basics of coding theory where we give general information and definition about coding theory. In chapter three and four, we define additive and additive codes for and a prime , which are natural generalizations of additive codes respectively. We determine the standard form of the generator and parity-check matrices of these codes and also we give some bounds on these codes and give examples that attain these bounds. In chapter five, we define linear codes where and is a ring with four elements. Although the structure of these codes and additive codes are similar, linear codes have advantages in some cases. We also give standard form of the generator matrix of these codes and define the dual space for a linear code and further we determine the generator matrix of the dual code which is actually a parity-check matrix of the code. In the last part of this chapter, we give some examples of these codes which are optimal binary codes under the Gray image. We also establish a MacWilliams-type identity between the weight distribution of a code and the weight distribution of a dual code. The last part which is the sixth chapter of the thesis is the conclusion part.