Asymptotically Tight MLD Bounds and Minimum-Variance Importance Sampling Estimator for Linear Block Codes Over BSCs

In this paper, we re-examine the classical problem of efficiently evaluating the block and bit error rate performance of linear block codes over binary symmetric channels (BSCs). In communication systems, the maximum likelihood decoding (MLD) bounds are powerful tools to predict the error performance of the coded systems, especially in the asymptotic regime of low error probability (or high signal-to-noise ratio). Contrary to the conventional wisdom, we prove that for BSCs, all bounds based on Gallager’s first bounding technique, including the famous union bound, are not asymptotically tight for all possible choices of the Gallager region. By proposing the so-called input demodulated-output weight enumerating function (IDWEF) of a code, asymptotically tight MLD upper and lower bounds for BSCs are then derived. In many practical scenarios where performance bounds are not applicable (e.g., due to the unavailability of the relevant coding parameters under a given decoder), the Monte Carlo simulation is commonly used despite its inefficiency, especially in the low error probability regime. We propose an efficient importance sampling (IS) estimator by deriving the optimal IS distribution of the Hamming weight of the error vector. In addition, the asymptotic relative saving on the required sample size of the proposed IS estimator over the state-of-the-art counterpart in the recent literature is characterized. Its accuracy in predicting the efficiency of the proposed IS estimator is verified by extensive computer simulation.