Foundations of information theory for coding theory

TL;DR

This paper introduces the mathematical foundations of information theory with a focus on its application to algebraic coding theory, covering key concepts like entropy, mutual information, and channel capacity, and explaining their relevance to coding.

Contribution

It bridges probabilistic information theory with algebraic coding techniques, emphasizing the theoretical limits of reliable communication over noisy channels.

Findings

Explains Shannon's information measures and their relevance to coding.

Describes the principles of maximum likelihood decoding.

Details Shannon's noisy channel coding theorem.

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.