Equivalent-Constraint Message Passing

Integrate this at the graph-construction and message-passing layers of a neural BP or GNN decoder. Start with a binary parity-check matrix H of shape m by n and measured syndrome s of length m. Construct a sparse binary matrix A with m_aug rows: retain all original identity rows, then add rows with Hamming weight two or three, such as e_i + e_j, provided the resulting check row A_k H does not duplicate an existing row. Compute H_aug = (A H) mod 2 and s_aug = (A s) mod 2. Reject candidate rows that produce zero or excessively dense checks, and cap the augmented variable and check degrees. Initialize variable embeddings from channel LLRs and check embeddings from s_aug. In each iteration, aggregate only along nonzero entries of H_aug using shared edge MLPs or learned scalar message scales. After T iterations, predict each variable error and optionally add a differentiable syndrome-consistency penalty. Pseudocode: sample candidate A; form H_aug and s_aug; initialize x and c; for t=1..T update checks then variables on H_aug; decode; optimize cross-entropy plus lambda times soft syndrome loss. Compute from the paper the matrix transformation and algebraic equivalence; estimate empirically 4-cycle count, degree growth, and oversquashing. Test on small LDPC codes with injected 4-cycles against the original neural BP graph. Success is lower error at equal message-passing steps or equal error with fewer steps, without more than about 2x graph density.