  We provide matching upper and lower bounds on the mutual information in noisy
reconstruction of parity check codes and thereby prove a long-standing
conjecture by Montanari [IEEE Transactions on Information Theory 2005]. Besides
extending a prior concentration result of Abbe and Montanari [Theory of
Computing 2015] to the case of odd check degrees, we precisely determine the
conjectured formula for code ensembles of arbitrary degree distribution, thus
capturing a broad class of capacity approaching codes.
