Geometrical and performance analysis of GMD and chase decoding algorithms

The overall number of nearest neighbors in bounded distance decoding (BDD) algorithms is given by N_o,eff = N_o + N_BDD, where N_BDD denotes the number of additional, noncodeward, neighbors that are generated during the (suboptimal) decoding process. We identify and enumerate the nearest neighbors associated with the original Generalized Minimum Distance (GMD) and Chase decoding algorithms. After careful examination of the decision regions of these algorithms, we derive an approximated probability ratio between the error contribution of a noncodeword neighbor (one of N_BDD points) and a codeword nearest neighbor. For Chase Algorithm 1 it is shown that the contribution to error probability of a noncodeword nearest neighbor is a factor of 2^d-1 less than the contribution of a codeword, while for Chase Algorithm 2 the factor is 2^[d/2]-1, d being the minimum Hamming distance of the code. For Chase Algorithm 3 and GMD, a recursive procedure for calculating this ratio, which turns out to be nonexponential in d, is presented. This procedure can also be used for specifically identifying the error patterns associated with Chase Algorithm 3 and GMD. Utilizing the probability ratio, we propose an improved approximated upper bound on the probability of error based on the union bound approach. Simulation results are given to demonstrate and support the analytical derivations.