Embedded Neural Belief Propagation: More Dimensions, Better Beliefs

Recently, Transformer-based decoders have achieved state-of-the-art performance on short codes, surpassing conventional belief propagation (BP) decoders. However, processing high-dimensional representations through attention incurs substantial computational cost. In this paper, we propose Embedded Neural Belief Propagation (E-NBP), which combines the efficiency of BP with the expressive power of high-dimensional representations. E-NBP embeds channel information into d-dimensional vectors and propagates message vectors along Tanner-graph edges. A key design choice is to perform intra-dimensional check-node updates to maintain low complexity, while performing cross-dimensional variable-node updates to improve decoding performance. On CCSDS(128, 64) at Eb/N0 = 6 dB, E-NBP reduces BER by 98.1% relative to CrossMPT while requiring 73.7% fewer FLOPs than EfficientMPT. These findings show that vector messages enable BP to compete with Transformer-based decoders while retaining its computational efficiency.