Curves in special position for algebraic geometric codes

Cyclic codes and Algebraic Geometric Codes (AG-Codes) are two important code families having strong algebraic structures. It is very easy to determine the minimum distances of cyclic codes and also the complexity of their decoding algorithms are very low. AGcodes, since they are linear codes obtained by evaluating function spaces on curves, it is possible to make sharp estimates about their minimum distances by using the theory of curves. Also because of their geometric structures, AG-codes have special decoding algorithms. Therefore, when studying a cyclic code, all the techniques of algebra and linear algebra, especially the properties of matrices of Vandermonde type; when studying an AG-code, besides of these techniques, very well developed devices of algebraic geometry, especially Riemann-Roch theory can be used. Moreover, since it is possible to de ne both families of codes by polynomials, the theory of Gr bner basis have very rich applications in both areas. By Saints and Heegard, the notion of AG-codes has been extended to include multi-dimensional cyclic codes, hyperbolic cascaded Reed-Solomon codes, ReedMuller codes and all AG-codes. Thereby, for both multi-dimensional cyclic codes and AGcodes, a uni ed theory has been revealed and with respect to this uni ed theory, a decoding algorithm technique has been given. But to apply this uni ed theory to an AG-code, the curve on which this AG-code de ned must be put into special position by a coordinate transformation. The importance of the curves in special position comes from the fact that in these cases it is possible to use Gr bner bases, because for a curve in special position all the computations can be done by polynomials instead of rational functions. In this thesis, rst of all, the basic topics of coding theory are mentioned; then, the basic notions about AG-codes are summarized. Subsequently, the topic of curves in special position for AG-codes are treated.