Foundations of information theory for coding theory
El Mahdi Mouloua, Essaid Mohamed
TL;DR
This note surveys foundational concepts in information theory with an eye toward algebraic coding theory. It presents entropy, conditional entropy, mutual information, and channel capacity, and illustrates them through the binary symmetric channel. It introduces maximum likelihood decoding and states Shannon's noisy channel coding theorem as the theoretical limit for reliable communication. By linking probabilistic information measures to coding-theoretic constructs, it provides a bridge between information theory and algebraic coding techniques.
Abstract
Information theory is introduced in this lecture note with a particular emphasis on its relevance to algebraic coding theory. The document develops the mathematical foundations for quantifying uncertainty and information transmission by building upon Shannon's pioneering formulation of information, entropy, and channel capacity. Examples, including the binary symmetric channel, illustrate key concepts such as entropy, conditional entropy, mutual information, and the noisy channel model. Furthermore, the note describes the principles of maximum likelihood decoding and Shannon's noisy channel coding theorem, which characterizes the theoretical limits of reliable communication over noisy channels. Students and researchers seeking a connection between probabilistic frameworks of information theory and structural and algebraic techniques used in modern coding theory will find this work helpful.