Generalized minimum-distance decoding in Euclidean-space: performance analysis

Detailed geometric analysis of decoding regions for GMD decoding is presented. We show that GMD decoding regions are non-convex in essentially all cases of interest. We also prove that these regions are always bounded by hyperplane segments. For d-erasure GMD decoding, we establish the presence of a large number of bounding hyperplanes at distance ⩽d+1. These results invalidate the estimates of performance derived from the union bound in the case of (multistage) GMD decoding. Alternative probabilistic estimates of, and upper bounds upon, the performance of GMD decoding are developed. Simulation results, for both low-dimensional and high-dimensional sphere packings, are in remarkably close agreement with our probabilistic approximations