Generalized minimum distance decoding in Euclidean space: Performance analysis

We present a detailed analysis of generalized minimum distance (GMD) decodi ng algorithms for Euclidean-space codes. In particular, we completely chara cterize GMD decoding regions in terms of receiver front-end properties, Thi s characterization is used to show that GMD decoding regions have intricate geometry. We prove that although these decoding regions are polyhedral; th ey 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 deco ding region, and thus do not pro,ide a meaningful measure of performance, A s an alternative, probabilistic estimates of, and upper bounds upon, the pe rformance 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 wit h multistage GMD decoding provide significant coding gains at a very law co mplexity. Simulated performance, in both cases, is in remarkably close agre ement with our probabilistic approximations.