  We consider near maximum-likelihood (ML) decoding of short linear block codes
based on neural belief propagation (BP) decoding recently introduced by
Nachmani et al.. While this method significantly outperforms conventional BP
decoding, the underlying parity-check matrix may still limit the overall
performance. In this paper, we introduce a method to tailor an overcomplete
parity-check matrix to (neural) BP decoding using machine learning. We consider
the weights in the Tanner graph as an indication of the importance of the
connected check nodes (CNs) to decoding and use them to prune unimportant CNs.
As the pruning is not tied over iterations, the final decoder uses a different
parity-check matrix in each iteration. For Reed-Muller and short low-density
parity-check codes, we achieve performance within 0.27 dB and 1.5 dB of the ML
performance while reducing the complexity of the decoder.
