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.