Abstract:
We show that for the case of the binary-symmetric channel and Gallager's decoding algorithm A the threshold can, in many cases, be determined analytically. More precisely, we show that the threshold is always upper-bounded by the minimum of 1 - λ2ρ′(1 -λ′(1)ρ′(1) - λ2ρ′(1) and the smallest positive real root τ of a specific polynomial p(x) and we observe that for most cases this bound is tight, i.e., it determines the threshold exactly. We also present optimal degree distributions for a large range of rates. In the case of rate one-half codes, for example, the threshold x* 0 of the optimal degree distribution is given by x* 0 ∼ 0.0513663. Finally, we outline how thresholds of more complicated decoders might be determined analytically. © 2004 IEEE.