A linear (error-correcting) code is a subspace of a vector space defined over a finite field. The basic parameters are the length or dimension of the full space, the rate or dimension of the subspace, and the minimum distance, which is the minimal Hamming weight of a vector in the subspace. Interesting codes come from various families.

Goppa’s use of algebraic geometry to analyze codes constructed from algebraic curves helped move the latter subject into mainstream applied mathematics. (These are often called algebraic-geometric codes even though there are many families of codes constructed using the techniques of algebraic geometry on varieties more general than curves.)

This paper, written for experts in algebraic-geometric codes, considers codes constructed from a tower of function fields, that is, a recursively defined sequence of fields F_i+i ^F_i; one gets curves in the standard way from these function fields. Garcia and Stichtenoth [1] found towers that were optimal with respect to the asymptotic ratio of the number of “points” on the curve to its genus (the number of points determines thelength of the code). The tower in this paper is not good in this sense, but it is good in the sense that the asymptotic ratio of the minimum distance to the length is positive. The treatment is quite complete, including explicit bases for the Riemann-Roch space that defines the code, and parameters of the dual codes. Many of the techniques are from the groundbreaking papers of Garcia and Stichtenoth.