Odd, even and repetitive occurrences codes on finite rings

The main purpose of today's communication processes is to ensure that the message to be transmitted is transmitted accurately, effectively and reliably. This process includes not only the transmission of information from one point to another, but also the retrieval and use of stored information at an appropriate time. In noisy and similar communication channels, it is of great importance to reduce errors that may occur during data transmission and to ensure reliable transmission. At this stage, the coding theory developed by scientists comes into play. This theory deals with issues such as detecting and correcting distortions that occur in communication errors and ensuring the accuracy of data. Algebraic definitions and basic information for coding theory are presented. The linear code definition is made and the properties of these codes such as minimum distance (minimum weight) are examined. Let C be code set, The mathematical foundations of the coding structure based on C the code are mentioned and various examples are shown. In addition, information about Hadamard codes is given in this section. In the ring (F_2 [v])⁄⟨v^2 - k⟩ , the case of v^2=1 is discussed in case k = 1. In this context, the F_2+vF_2 ring was examined and the structure created by addition and multiplication operations in this ring was presented in a detailed table. Ideals of the ring were determined and the connections between them were explained. In addition, the relationship of the codes defined in this ring with Hadamard codes and their connection with quasi-cyclic codes was shown. The first and second repetitive occurrences code definitions were made on the F_2+vF_2 ring and examples were given. Odd and even codes are defined and examples were presented. In the case of k = v, studies were made on the (F_2 [v])⁄⟨v^2 - k⟩ ring, and for the case of v^2=v, studies were made on the F_2+vF_2 ring. In this case, the relationship between cyclic codes and quasi-cyclic codes was given. In order to determine the code parameters, the minimum distance function and the weight function (distance) in the F_2+vF_2 ring were examined and the Hamming weight was defined. The general Gray transformation that gives the relationship between the ring and the object was shown. The first and second repetitive occurrences code definitions were made on this ring and examples were given. In addition, odd and even codes are defined and examples are shown. Gray transforms of these codes are taken and their relations with other codes are given. In the case of v^3=0, codes are obtained based on special matrices for the eight-element ring Z_2+vZ_2+v^2 Z_2 and the parameters of these codes are calculated. Using these parameters, the first repetitive occurrences code and the second repetitive occurrences code are defined and the parameters of the Gray images of these codes are found and examples are shown.