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.