Error Correcting Codes

Overview

Course Overview & Basics - Example Codes and their Parameters - Mathematical Preliminaries: Groups - Subgroups and Equivalence Relations - Cosets, Rings & Fields - Vector Spaces, Linear - Linear Codes, & Linear independence - Spanning & Basis - The Dual Code - Systematic Generator Matrix - Minimum Distance of a Linear Code - Bounds on the size of a Code - Asymptotic Bounds - Standard Array Decoding - Performance Analysis of the SAD - State and Trellis - The Viterbi Decoder - Catastrophic Error Propagation - Path Enumeration - Viterbi Decoder over the AWGN Channel - Generalized Distributive Law - The MPF Problem - Further Examples of the MPF Problem - Junction Trees recap - Example of Junction Tree Construction

Message passing on the Junction tree - GDL Approach to Decoding Convolutional Codes - ML Code-Symbol Decoding of the Convolutional Code - LDPC Codes - LDPC Code Terminology - Gallager Decoding Algorithm A - BP Decoding of LDPC Codes - BP Decoding - Density Evolution under BP decoding - Convergence & Concentration Theorem -- LDPC Codes - A Construction for Finite Fields - Finite Fields: A Deductive Approach - Deductive Approach to Finite Fields - Subfields of a Finite field - Transform Approach to Cyclic Codes - Estimating the Parameters of a Cyclic Code - Decoding Cyclic Codes