Equidistance and unit codes

Abstract

In coding theory, single codes, as the implementation of non-traditional coding methods, are focused in particular on such a practical application as the transmission of symbols (bits) in communication channels. Today, the analysis and study of unit codes is a priority and actuality in terms of evaluating their corrective properties. This paper analyzes the properties of three unit codes, such as unit position (marking), unit pair and unit normal codes. The comparative characteristic of these codes taking into account their correcting properties is given. It is taken into account that the corrective properties of the code are determined by the code distance, which is the minimum distance between its code points. The formulas for determining the average probability of error non-detection for the received correction codes are given. Research and proof of corrective properties of unit codes were carried out from the point of view of their equidistance. Such codes are characterized by the fact that the code distance in the equidistant code must be an even number. The analysis of the characteristics of unit codes presented in this work showed that one of the optimal among equidistant unit codes can be considered a unit position (marking) code. The formula for calculating the lower estimate of the average probability of error non-detection for any probability of error-free transmission of the symbol, which coincides with the value of the average probability of error non-detection for the McDonald's equidistant code. This also confirmed the optimality of a considered unit position (marking) code. With this in mind, the application area of the unit position code, as noise immunity, extends due to the possibility of encoding the states of correcting machines and addressing data in computer storage devices.