Analytic Expressions of Decoding Thresholds for LDPC Codes Over BEC

Analytic expressions of decoding thresholds both for regular and irregular low-density parity-check (LDPC) codes over the binary erasure channel (BEC) are given, and low-complexity methods for determining the thresholds are proposed based on the analytic expressions. Specifically, firstly, a fixed-point equation, which represents a constraint relationship between the threshold and erasure probability of transmitted bits, is established. Secondly, by introducing an auxiliary equation, the threshold can be solved as an analytically function of solutions of a polynomial equation. Moreover, this method is extended to the Additive White Gaussian Noise (AWGN) channel. Numerical results show that, our proposed method is more accurate, and the complexity is much lower than that of conventional density evolution (DE). Moreover, our method does not require to compute the inverse of degree distribution function for irregular LDPC codes.