A lower bound on the error rate of linear binary block codes (under maximum likelihood decoding) over BSC channels is proposed. According to the principle of the maximum likelihood (ML) decoding algorithm, the decoding error probability is firstly converted into the joint probability of the error events, and the judge rule of the redundant error events is deduced based on the optimization rule of the improved Dawson-Sankoff bound. Moreover, the calculation expression about lower bound of the error probability solely depends on the Hamming weight enumerator function of the code and the crossover probability of the channel. The simulation results applying to various LDPC codes show that the new lower bound outperforms those generic lower bounds and the sphere packing bound. Its computational complexity is also lower.