A lower bound on the error rate of linear binary block codes (under maximum likelihood decoding) over BPSK-modulated AWGN channels is proposed in the current study. The essence of the optimization to the lower error rate bound is to optimize the lower bound on the union probability of finite events. Combined with the optimizing rule of the improved Dawson-sankoff bound, we obtain the judge rule of the redundant error events and obtain the calculation expression about lower bound of SER which solely depends on the Hamming weight enumerator function of the code and the signal-to-noise ratio (Eb/N0). The algorithm can get the tighter lower bound and has lower complexity than the de Caen's kind bound and the KAT kind bound, which has been proved by the numerical results applying to LDPC codes and the others.