Vector bundles and codes on algebraic curves

The construction of algebraic-geometric (AG) codes can be seen in various ways as a geometric process, but their known decoding algorithms rely on algebraic ideas that have no direct geometric interpretation. The goal of this dissertation is to make the connection between corrupted codewords of an AG code and the geometry of the underlying curve more explicit.; The starting point is a known (highly abstract) interpretation of decoding in the language of vector bundles: the syndrome of an error vector is identified with an extension of two fixed line bundles, and the error locations can be obtained (at least in principle) from its minimal quotient bundle. We show how to transform a correctable word into a transition matrix for the associated vector bundle. This involves finding functions on the curve that satisfy a certain property in the coefficients of their power series expansions around a distinguished point, computing the images of the corresponding global sections of a line bundle under Serre duality, and specifying a suitable open cover of the curve.; Improved constructions are given for a widely studied class of codes on the Hermitian curve. The improved cover, for example, allows any rational point to be expressed as a line bundle by a simple kind of transition function.; We also describe the structure of a vector bundle as an abstract algebraic variety in order to determine the concrete form of an embedding of bundles. This allows the decoding problem for AG codes to be viewed as a search for a certain combination of rational functions on the curve.; Finally, we consider the problem of constructing rank two vector bundles with exceptionally many maximal subbundles defined over a finite field. A connection to an open question of coding theory is observed, which suggests a number of concrete research problems of general interest.