6 - Applications to Algebraic Coding Theory

Summary

Goppa's celebrated construction of algebraic-geometry codes uses algebraic curves over finite fields with many rational points or, equivalently, global function fields with many rational places. This construction was a breakthrough in algebraic coding theory because it yields sequences of linear codes beating the asymptotic Gilbert-Varshamov bound. We describe this construction and its consequences, but also recent work which shows that improvements on Goppa's construction can be obtained by other constructions that also employ places of higher degree. As basic references for algebraic coding theory we recommend the books of Mac Williams and Sloane [77] and van Lint [166].

Goppa's Algebraic-Geometry Codes

We start with a brief recapitulation of the theory of linear codes. Recall that a code is a scheme for detecting and correcting transmission errors in noisy communication channels. A code operates by adding redundant information to messages. As the signal alphabet we always use F₉, where q = 2 is naturally an important special case.

A linear code over F_q is a nonzero linear subspace of the vector space for some n ≥ 1. If C ⊆ is a linear code over F_q, then n is the length of C and k≔ dim(C) is the dimension of C. We express these facts by saying that C is a linear [n, k] code over F_q.