Applications of Error-Correcting Codes

Abstract

Coding theory, a branch of mathematics, focuses on the detection and correction of errors in data to ensure reliable transmission through noisy channels. Originating from Shannon's 1948 seminal work, it plays a crucial role in digital communication, impacting fields ranging from deep-space transmissions to digital data storage. This dissertation explores the foundations and advanced concepts of error-correcting codes, with an emphasis on binary and linear codes, finite fields, and their applications. Key topics include encoder and decoder models, finite field arithmetic, Hamming and cyclic codes, and practical applications such as ISBN and data reliability systems. The research demonstrates the power of coding theory in mitigating noise during transmission, ensuring data integrity, and guiding the future development of communication technologies. The role of computational tools, such as Matlab, in analyzing and implementing coding schemes, is also examined.