Low-Pathwidth GRAND: Exact Likelihood-Ordered Enumeration for BPSK Transmission over Correlated Gaussian Noise

Abstract

The finite-block maximum-likelihood (ML) guarantee of soft-input GRAND requires querying noise-effect patterns in nonincreasing conditional-likelihood order. Under correlated Gaussian noise, additive reliability metrics and independent-block approximations need not preserve this order because the matched metric contains cross-coordinate interactions; the first codebook hit need not induce an ML codeword. We develop Low-Pathwidth GRAND (LP-GRAND) for binary phase-shift keying (BPSK) with precision matrix $Q$. The candidate-dependent part of the Gaussian negative log-likelihood is an observation-dependent quadratic pseudo-Boolean energy whose interaction graph has edge $\{i,j\}$ exactly when $Q_{ij}\ne 0$. If $Q$ has half-bandwidth at most $\hat{I}½$, this energy admits a trellis with at most $2^{\hat{I}}½$ states per layer; a path decomposition of width $w$ yields at most $2^{w+1}$ bag assignments per layer. In real arithmetic, suffix dynamic programming and best-first complete-path enumeration enumerate patterns in nondecreasing energy. With complete enumeration and no abandonment, the first codebook hit induces an ML codeword for any nonempty binary codebook with equiprobable codewords. LP-GRAND agreed with exhaustive codeword ML in all $10,000$ frames for two $[20,12]$ codes. At nominal $E_b/N_0=2$ dB, its empirical BLER was lower than that of each block-based approximation for six $[64,52]$ codes.

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