Asymptotic and finite-length performances

Note that the symmetry property implies that the channel output coincides with the log-likelihood ratio `(y) = y. By this correspondence, symmetric densities admit a partial ordering by physical degradation Z1 Z2 if the corresponding chanels are ordered in the same way Q1 Q2 (i.e. Q1 is less noisy than Q2). Consider now belief propagation decoding with messages {hi→a}, {ĥa→i} and soft decisions {hi}, all of these in the log-likelihood ratio domain. We saw that –as n→∞– these converge to limit random variables described by density evolution