A brief guide to Goppa codes

Goppa codes, or algebraic geometry codes (also called AG codes), were discovered by V. D. Goppa in the early 1980's [Go]. Other Russian mathematicians contributed to their theory, such as Yu. I. Manin, M. A. Tsfasman, and S. G. Vladut. Certain Goppa codes arising from ``modular curves'' gave the first infinite family of codes whose parameters beat the Gilbert-Varshamov bound. This was quite an interesting result since before then no infinite familty of codes were known having this property. In this note we sketch the terrain of AG codes arising from curves over finite fields using MAGMA [MAGMA]. There are two types of Goppa codes: ``function codes'' and ``primary (or residue) codes''. One is equivalent to the dual code of the other, so shall only go into some detail about one of them. Each of them requires a certain amount of data to begin: one must

• choose a finite field $\mathbb{F}$,
• choose a smooth algebraic curve $C$ over $\mathbb{F}$,
• pick rational points $P_1,P_2,...,P_n$ in $C(\mathbb{F})$,
• choose a (not necessarily rational) divisor $D$ distinct from the $P_i$'s,
• determine a basis for the ``Riemann-Roch space'' $L(D)$.

Undefined are ``smooth algebraic curve'', ``rational points'', ``divisors'', ``Riemann-Roch space''. A divisor is simply a (formal) finite sum of points of $C(\overline{\mathbb{F}})$, where $\overline{\mathbb{F}}$ is an algebraic closure of $\mathbb{F}$ (see Definition 2.1.7) and $C(\overline{\mathbb{F}})$ is defined below. The other terms are more technical to define and we shall stick to examples and how to do computations with the (using [MAGMA]) rather that formal definitions. For more precise details, see Pretzel's book [P]. Below, we shall focus on what is needed to work out examples using MAGMA.