An Analytical Study: The Mathematical Relationship Between Hamming Theory for Error Correction and Representation in Classical Spaces with Applications of Supervised Machine Learning

Abstract

This paper presents a comprehensive theoretical and empirical analysis of the mathematical relationship between Hamming code error-correction mechanisms and their geometric representation within Euclidean vector spaces, with particular emphasis on supervised machine learning applications. Recent developments in deep learning frameworks have demonstrated significant potential for enhancing decoding procedures beyond traditional algorithmic approaches. Our research combines rigorous theoretical foundations with experimental validation, achieving 100% error-correction accuracy for the (7,4) Hamming code through a carefully designed multi-layer neural network architecture by MATLAB. The study reveals that neural-network-based decoders significantly outperform conventional algorithms in terms of accuracy, robustness, and noise tolerance when operating in challenging environments. These findings contribute to the growing body of knowledge at the intersection of coding theory, vector space mathematics, and artificial intelligence, providing both theoretical insights and practical implications for next-generation communication systems.