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Algebraic and geometric fundamentals of geometrically uniform codes

Abstract

The research in algebraic and geometric fundamentals of geometrically uniform codes aims at introducing topological structures in the encoding and decoding processes not only in the traditional algebraic structure of fields as well as in the algebraic structures of rings and groups. Hence, the design of new signal constellations and of new classes of codes by studying the topological spaces, metric spaces, and the corresponding surfaces where they are embedded bring new light to the current approach. As an example, certain source and channel coding problems can be viewed as one through the existence of a homeomorphism between the corresponding metric spaces. Associated with this approach, it is our aim to propose coding techniques for linear q-ary codes, G-linear codes, where G is a group, convolutional codes over groups, characterize the surfaces of linear trellis codes, to propose new signal constellations and corresponding demodulations in hyperbolic geometry, codes over algebraic integers, etc. Since the computational complexity of the problems involved in this research is high we also intend to develop the corresponding software. (AU)

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