Improved Lower Bounds for the Error Rate of Linear Block Codes

Abstract We obtain two lower bounds on the error rate of linear binary block codes (under maximum likelihood decoding) over BPSK-modulated AWGN channels. We cast the problem of finding a lower bound on the probability of a union as an optimization problem which seeks to find the subset which maximizes a recent lower bound – due to Kuai, Alajaji, and Takahara – that we will refer to as the KAT bound. Two variations of the KAT lower bound are then derived. The first bound, the LB-f bound, requires the weight of the product of the codewords with minimum weight in addition to their weight enumeration, while the other bound, the LB-s bound (which is the main contribution of this paper), is algorithmic and only needs the weight enumeration function of the code. The use of a subset of the codebook to evaluate the KAT lower bound not only reduces computational complexity, but also tightens this bound specially at low signal-to-noise (SNR) ratios. Numerical results for binary block codes indicate that at low SNRs the LB-f bound is tighter than the LB-s bound. At high SNRs, the LB-s bound is tighter than other recent lower bounds in the literature, which comprise the lower bound due to Séguin, the original KAT bound (evaluated on the entire codebook), and the dot-product and norm bounds due to Cohen and Merhav.