An Efficient Algorithmic Lower Bound for the Error Rate of Linear Block Codes

We present an efficient algorithmic lower bound for the block error rate of linear binary block codes under soft maximum-likelihood decoding over binary phase-shift keying modulated additive white Gaussian noise channels. We cast the problem of finding a lower bound on the probability of a union as an optimization problem that seeks to find the subset that maximizes a recent lower bound - due to Kuai, Alajaji, and Takahara - that we will refer to as the KAT bound. The improved bound, which is denoted by LB-s, is asymptotically tight [as the signal-to-noise ratio (SNR) grows to infinity] and depends only on the code's weight enumeration function for its calculation. The use of a subset of the codebook to evaluate the LB-s lower bound not only significantly reduces computational complexity, but also tightens the bound specially at low SNRs. Numerical results for binary block codes indicate that at high SNRs, the LB-s bound is tighter than other recent lower bounds in the literature, which comprise the lower bound due to Seguin, the KAT bound (evaluated on the entire codebook), and the dot-product and norm bounds due to Cohen and Merhav.