Explore an MAA Invited Address on coding theory and finite fields in this 54-minute conference talk. Dive into the fascinating world of codes derived from polynomials over finite fields, covering topics such as Reed-Solomon codes, MDS codes, and Reed-Muller codes. Learn about communication over noisy channels, the main problem in combinatorial coding theory, and the Hamming weight enumerator. Discover the connections between algebraic geometry and coding theory, including codes from cubic and quartic curves. Examine the MacWilliams identity and its implications for dual codes. Gain insights into the practical applications of these mathematical concepts in modern communication systems.
Overview
Syllabus
Intro
Communication over a Noisy Channel
Coding Theory Basics
Main Problem in Combinatorial Coding Theory
Tables for Linear Codes (codetables.de)
Reed-Solomon Codes are MDS
Doubly Extended (Projective) Reed-Solomon Codes
Reed-Solomon Code: Example 2
Main Conjecture for MDS Codes Ill
More Variables. Reed-Muller Codes
Reed-Muller Codes II
The Hamming Weight Enumerator of a Code Definition
Quadratic Polynomials in 2 Variables
Reed-Muller Codes from Cubic Curves
Reed-Muller Codes from Quartic Curves
Rational Point Counts for Quartic Curves: Asymmetry Definition
The Dual Code of a Linear Code
The MacWilliams Identity
Algebraic Geometry Codes
Codes to Communication
Taught by
Joint Mathematics Meetings
