ПОБУДОВА КОДІВ ХЕММІНГА В СКІНЧЕННИХ ПОЛЯХ ГАЛУА

Abstract

Hamming codes and their modifications are key in numerous technological processes and systems, where minimizing errors is crucial for enhancing reliability and efficiency in data transmission or storage. They allow for the detection and automatic correction of single errors in each data block. Their relative simplicity to implement at the hardware and software levels, renders them useful in systems demanding high reliability. Moreover, they optimize transmission channel use due to their minimal redundancy. At work is the task of applying the properties of modular arithmetic to develop codes that can effectively correct errors in systems with higher bit rates than binary has been considered. Conducted research confirms that such codes can be tailored to meet specific data transmission system requirements, showcasing adaptability. The use of Hamming codes within finite Galois fields, leveraging modular arithmetic, further enhances correction capabilities and efficiency, streamlining implementation and computation processes. Error correction codes in finite Galois fields use the same principles applied in Hamming codes but take into account the mathematical properties of these fields for effective encoding, detection, and correction of errors.The proposed error correction method, through integration with modular arithmetic, opens possibilities for optimizing encoding and decoding processes, allowing for a higher level of data transmission reliability with minimal redundancy, which was previously limited by the properties of traditional codes. The proposed solution demonstrates the potential for using modular arithmetic for error correction but can provide a basis for further research in this area, opening new possibilities for improving information processing and transmission technologies.