  We revisit recent methods that employ graph neural networks for decoding
error correcting codes and employ messages that are computed in an
autoregressive manner. The outgoing messages of the variable nodes are
conditioned not only on the incoming messages, but also on an estimation of the
SNR and on the inferred codeword and on two downstream computations: (i) an
extended vector of parity check outcomes, (ii) the mismatch between the
inferred codeword and the re-encoding of the information bits of this codeword.
Unlike most learned methods in the field, our method violates the symmetry
conditions that enable the other methods to train exclusively with the
zero-word. Despite not having the luxury of training on a single word, and the
inability to train on more than a small fraction of the relevant sample space,
we demonstrate effective training. The new method obtains a bit error rate that
outperforms the latest methods by a sizable margin.
