Vector bundles and codes on the Hermitian curve

The construction of algebraic-geometry (AG) codes can be seen as a distinctly geometric process, and yet decoding procedures tend to rely on algebraic ideas that have no direct geometric interpretation. Recently, however, Trygve Johnsen observed that decoding can be viewed in abstract terms of a class of vector bundles on the underlying curve. The present paper describes these objects at a concrete computational level for the Hermitian codes C/sub /spl Omega//(D,mP/sub /spl infin//) defined over F/sub q//sup 2/ (q a power of 2). The construction of explicit representations of the vector bundles by transition matrices involves finding functions on the curve that satisfy a certain property in their power series expansions around P/sub /spl infin//, computing the image of the corresponding global sections under Serre duality, and finding a suitable open cover of the curve. The cover enables any rational point to be expressed as a line bundle by a simple kind of transition function. A special case is considered in which these functions can be realized as ratios of linear forms.