New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes

In this paper, new techniques are presented to either simplify or improve most existing upper bounds on the maximum-likelihood (ML) decoding performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectrum by Ma et al. is re-derived in a detailed way based on Gallager's first bounding technique (GFBT), where the "good region" is specified by a sub-optimal list decoding algorithm. The error probability caused by the bad region can be upper-bounded by the tail-probability of a binomial distribution, while the error probability caused by the good region can be upper-bounded by most existing techniques. Secondly, we propose two techniques to tighten the union bound on the error probability caused by the good region. The first technique is based on pair-wise error probabilities. The second technique is based on triplet-wise error probabilities, which can be upper-bounded by the fact that any three bipolar vectors form a non-obtuse triangle. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the Q-function. The proposed bounds can also be adapted to bit-error probabilities.