Algebra in Algorithmic Coding Theory
Madhu Sudan
TL;DR
This survey traces how algebraic methods underpin both code design and efficient decoding, culminating in capacity-achieving schemes like Folded-Reed-Solomon codes. It details RS list-decoding via interpolation with Q(X,Y), refinements using weighted degrees and multiplicities, and the pivotal role of bundling (FRS) to reach capacity, including rate-reducing reductions that preserve decodability. The discussion surveys alphabet-size tradeoffs, decoding runtimes, and a spectrum of capacity-achieving codes beyond RS, while highlighting open problems in binary-capacity and explicit constructions. Overall, the work demonstrates how relatively elementary algebra over finite fields yields powerful, near-optimal coding procedures with practical decoding algorithms.
Abstract
We survey the notion and history of error-correcting codes and the algorithms needed to make them effective in information transmission. We then give some basic as well as more modern constructions of, and algorithms for, error-correcting codes that depend on relatively simple elements of applied algebra. While the role of algebra in the constructions of codes has been widely acknowledged in texts and other writings, the role in the design of algorithms is often less widely understood, and this survey hopes to reduce this difference to some extent.