  In this paper, the sphere bound (SB) is revisited within a general bounding
framework based on nested Gallager regions. The equivalence is revealed between
the SB proposed by Herzberg and Poltyrev and the SB proposed by Kasami et al.,
whereas the latter was rarely cited in the literatures. Interestingly and
importantly, the derivation of the SB based on nested Gallager regions suggests
us a new simulation approach to performance evaluation of binary linear codes
over additive white Gaussian noise (AWGN) channels. In order for the
performance evaluation, the proposed approach decouples the geometrical
structure of the code from the noise statistics. The former specifies the
conditional error probabilities, which are independent of signal-to-noise
ratios (SNRs) and can be simulated and estimated efficiently, while the latter
determines the probabilities of those conditions, which involve SNRs and can be
calculated numerically. Numerical results show that the proposed simulation
approach matches well with the traditional simulation approach in the high
error rate region but is able to evaluate efficiently the performance in the
extremely low error rate region.
