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In this paper, the sphere bound (SB) is revisited within a general boundingframework based on nested Gallager regions. The equivalence is revealed betweenthe SB proposed by Herzberg and Poltyrev and the SB proposed by Kasami et al.,whereas the latter was rarely cited in the literatures. Interestingly andimportantly, the derivation of the SB based on nested Gallager regions suggestsus a new simulation approach to performance evaluation of binary linear codesover additive white Gaussian noise (AWGN) channels. In order for theperformance evaluation, the proposed approach decouples the geometricalstructure of the code from the noise statistics. The former specifies theconditional error probabilities, which are independent of signal-to-noiseratios (SNRs) and can be simulated and estimated efficiently, while the latterdetermines the probabilities of those conditions, which involve SNRs and can becalculated numerically. Numerical results show that the proposed simulationapproach matches well with the traditional simulation approach in the higherror rate region but is able to evaluate efficiently the performance in theextremely low error rate region.

A New Simulation Approach to Performance Evaluation of Binary Linear Codes in the Extremely Low Error Rate Region