Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Grobner bases

In this paper, it is proved that any algehraicgeometr ic code can be expressed as a cross section of an extended multidimensional cyclic code. Both algebraic-geometric codes and multidimensional cyclic codes are descr ibed by a unified theory of l inear block codes def ined over point sets: algebraic-geometric codes are def ined over the points of an algebraic curve, and an m-dimensional cyclic code is def ined over the points in mdimensional space. The power of the unified theory is in its description of decoding techniques using GrSbner bases. In order to fit an algebraic-geometric code into this theory, a change of coordinates must be appl ied to the curve over which the code is def ined so that the curve is in special position. For curves in special position, all computat ions can be performed with polynomials, rather than rational functions, and this also makes it possible to take advantage of the theory of Grobner bases. Next, a transform is def ined for algebraic-geometric codes which general izes the discrete Fourier transform. The transform is also related to a GrSbner basis, and is useful in setting up the decoding problem. In the decoding problem, a key step is finding a GrSbner basis for an error locator ideal. For algebraic-geometric codes, multidimensional cyclic codes, and indeed, any cross section of an extended multidimensional cyclic code, Sakata’s algorithm can be used to find linear recursion relations which hold on the syndrome array. In this general context, we give a self-contained and simplified presentat ion of Sakata’s algorithm, and present a general f ramework for decoding algorithms for this family of codes, in which the use of Sakata’s algorithm is supplemented by a procedure for extending the syndrome array.