Neural Decoding with Redundant and Reliability-Conditioned Dual Relations

Neural decoders for linear codes typically treat the supplied parity-check matrix as a fixed structural description, even though a code admits many algebraically equivalent full-rank bases of the same dual space. We show that this choice is not neutral for finite neural decoders: equivalent parity-check bases can change bit-error rate (BER) by more than $7\times$ under matched code, rank, row space, architecture, and parameter count, with a residual gap remaining even under an exact row-weight-matched control. We then move beyond a single basis by exposing a rank-complete vocabulary of redundant dual relations and conditioning each relation on a normalized log soft-parity confidence derived from the current observation. Across Gallager LDPC, BCH, and MacKay LDPC, moving beyond repeated native parity coordinates to redundant dual-relation representations consistently improves a fixed CrossMPT backbone, while the benefit of particular relation sets is code dependent. Relation-level reliability provides an additional code-dependent gain and remains effective in a larger-blocklength replication. On MacKay LDPC, reassigning the same per-frame reliability values to incorrect equal-weight relation identities increases BER by roughly $51$–$94\%$ across 3–5 dB, and paired rescue–harm analysis shows that reliability conditioning more often repairs than introduces frame errors. Together, these results show that neural decoding depends not only on which parity relations are exposed, but also on how their observation-dependent reliability is represented to the decoder.