Exact thresholds for low-density parity-check codes over the binary erasure channel

A simple method for determining the threshold of irregular LDPC codes over the binary erasure channel (BEC) under message-passing decoding is proposed. An exact formula for calculating the threshold of irregular LDPC codes over the BEC is proved. This generalizes the known result on the threshold of regular LDPC codes to irregular LDPC codes. Moreover, our new method can avoid the computation of the inverse of the degree distribution function for irregular LDPC codes. Numerical results demonstrate the correctness of our proposed method.