Generalized minimum distance decoding in Euclidean space: performance analysis

We present a detailed analysis of generalized minimum distance (GMD) decoding algorithms for Euclidean space codes. In particular, we completely characterize GMD decoding regions in terms of receiver front-end properties. This characterization is used to show that GMD decoding regions have intricate geometry. We prove that although these decoding regions are polyhedral, they are essentially always nonconvex. We furthermore show that conventional performance parameters, such as error-correction radius and effective error coefficient, do not capture the essential geometric features of a GMD decoding region, and thus do not provide a meaningful measure of performance. As an alternative, probabilistic estimates of, and upper bounds upon, the performance of GMD decoding are developed. Furthermore, extensive simulation results, for both low-dimensional and high-dimensional sphere-packings, are presented. These simulations show that multilevel codes in conjunction with multistage GMD decoding provide significant coding gains at a very low complexity. Simulated performance, in both cases, is in remarkably close agreement with our probabilistic approximations.