Coding Theory 00: Roadmap to AG Codes
Coding theory / 00
Roadmap to AG Codes
This article is the coverage audit and reading map for the companion series to Algebraic geometric codes on curves and surfaces.
The source text has a compact structure: elementary coding theory, algebraic-geometric codes on curves, decoding algorithms, scheme-theoretic preparation for higher-dimensional varieties, surface codes, and ruled surface codes. The purpose of this reorganized collection is to make that path readable as a sequence of lecture notes. The articles are a companion designed to supply the missing connective tissue, computations, notation, and examples.
Notation used throughout
The finite field is \(\mathbb F_q\). A linear code has parameters \([n,k,d]_q\). A curve is smooth, projective, geometrically integral, and defined over \(\mathbb F_q\) unless explicitly stated otherwise. For a divisor \(G\) on a curve, \(L(G)=\{f:(f)+G\ge0\}\cup\{0\}\). Evaluation codes are denoted \(C_L(D,G)\) and differential codes \(C_\Omega(D,G)\), with \(D=P_1+\cdots+P_n\) disjoint from \(G\).
Reading order
The first two articles reconstruct Chapter 1. The finite-field and polynomial-code bridge is made explicit through the Singleton-tight polynomial example, the syndrome examples over finite fields, and the Reed-Solomon realization on \(\mathbb P^1\) before the general curve construction. Articles 03-06 give the curve theory of Chapter 2. Articles 07-09 separate the decoding chapter into locators, majority voting, and worked examples. Articles 10-11 supply the scheme and sheaf language used in Chapter 4. Articles 12-13 cover surfaces. Articles 14-17 cover ruled surfaces. Article 18 records the open problems and the research map.