TY - JOUR

T1 - Geometrical and performance analysis of GMD and chase decoding algorithms

AU - Fishier, Eran

AU - Amrani, Ofer

AU - Be'ery, Yair

PY - 1999

Y1 - 1999

N2 - The overall number of nearest neighbors in bounded distance decoding (BDD) algorithms is given by No,eff = No + NBDD, where NBDD 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 NBDD 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 2d-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.

AB - The overall number of nearest neighbors in bounded distance decoding (BDD) algorithms is given by No,eff = No + NBDD, where NBDD 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 NBDD 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 2d-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.

UR - http://www.scopus.com/inward/record.url?scp=0032666997&partnerID=8YFLogxK

U2 - 10.1109/18.771143

DO - 10.1109/18.771143

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AN - SCOPUS:0032666997

VL - 45

SP - 1406

EP - 1422

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 5

ER -