Expander Graphs and Expander Codes: Foundations and Decoding

Expander graphs are fundamental structures in theoretical computer science, combinatorics, and modern coding theory, offering high connectivity with sparse edges. This course provides a clear, step-by-step introduction to graph expansion and its practical applications in error-correcting codes. You will begin by mastering essential definitions and properties before progressing to advanced topics like decoding algorithms and expansion coefficients. By reading focused explanations and walking through theoretical proofs, you will build a solid intuition for how combinatorial expanders work and why they are vital in distributed computing and error correction. What you'll learn: - Understand the core concepts and definitions of vertex expansion and edge expansion in graphs - Analyze combinatorial expanders and their characteristic expansion coefficients - Apply the unique neighbor lemma (isolated neighbors lemma) to establish expansion bounds - Explore the design principles of expander-based error-correcting codes - Trace parallel decoding algorithms operating in logarithmic iterations - Connect spectral and combinatorial properties to modern distributed and network algorithms The course begins with foundational graph theory terms and combinatorial basics, moves into expansion properties and analytical lemmas, and concludes with algorithmic implementations for error decoding. It is designed for learners, students, and engineers interested in discrete mathematics and coding theory, with no advanced prior knowledge required. Start reading today to deepen your understanding of expander graph theory.