Error correcting codes with algebraic decoding algorithms for multidimensional signals

Recently, some multiplicative groups of complex integers, i.e, Gaussian integers, were constructed to obtain quadrature amplitude modulation (QAM) signal spaces and to code the QAM signals such that a differentially coherent method can be applied to demodulate the QAM signals. It was proposed to find other similar and larger groups of complex integers without resorting to brute force computer calculations. On the other hand, for digital data transmission, signals over a two or multidimensional signal space are often selected for bandwidth efficiency. Although block codes over finite fields have an elegant algebraic theory, they are not suited for coding over two or multidimensional signal constellations. Most high performance communication systems use convolutional codes at least as inner codes. This is mainly due to the fact that there is efficient soft decision decoding algorithms which allow maximum likelihood decoders. This thesis studies error correcting codes with algebraic decoding algorithms for multidimensional signals. Block codes constructed in this thesis allow an algebraic approach in an area which is currently mainly dominated by nonalgebraic convolutional codes.Doctor of Philosophy (EEE