New techniques for upper-bounding the MLD performance of binary linear codes

In this paper, two techniques are presented to either simplify or improve most of the existing upper bounds on the maximum-likelihood decoding (MLD) performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectra by Ma et al is re-derived in a detailed way based on Gallager´s first bounding technique (GFBT). Secondly, we propose using triplet-wise error probabilities instead of pair-wise error probabilities to improve the union bound. In doing so, we prove that any three codewords form a non-obtuse triangle, which can be utilized to upper-bound the triplet-wise error probability. 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.