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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., Comment: arXiv admin note: text overlap with arXiv:1202.0592

A new Chase-type soft-decision decoding algorithm for Reed-Solomon codes is proposed, referred to as tree-based Chase-type algorithm}. The proposed tree-based Chase-type algorithm takes the set of all vectors as the set of testing patterns, and hence definitely delivers the most-likely codeword provided that the computational resources are allowed. All the testing patterns are arranged in an ordered rooted tree according to the likelihood bounds of the possibly generated codewords, which is an extension of Wu and Pados' method from binary into $q$-ary linear block codes. While performing the algorithm, the ordered rooted tree is constructed progressively by adding at most two leafs at each trial. The ordered tree naturally induces a sufficient condition for the most-likely codeword. That is, whenever the tree-based Chase-type algorithm exits before a preset maximum number of trials is reached, the output codeword must be the most-likely one. But, in fact, the algorithm can be terminated by setting a discrepancy threshold instead of a maximum number of trials. When the tree-based Chase-type algorithm is combined with Guruswami-Sudan (GS) algorithm, each trial can be implement in an extremely simple way by removing from the gradually updated Grobner basis one old point and interpolating one new point. Simulation results show that the tree-based Chase-type algorithm performs better than the recently proposed Chase-type algorithm by Bellorado et al with less trials (on average) given that the maximum number of trials is the same.