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

2026-07-30

AI Summary

This work addresses the limitation of conventional GRAND algorithms in correlated Gaussian noise, where ignoring the noise covariance structure compromises maximum-likelihood (ML) optimality. Focusing on BPSK modulation, the paper proposes a novel approach that integrates graph path decomposition with tree-based dynamic programming. By exploiting the sparsity of the precision matrix to construct low-treewidth decompositions and combining pseudo-Boolean quadratic energy modeling, suffix-based dynamic programming, and best-first search over the reals, the method enumerates noise effect patterns in exact likelihood order. This guarantees, for the first time, that the first codeword hit under any non-empty equiprobable binary codebook is the true ML solution. Experiments confirm perfect agreement with exhaustive ML over 10,000 frames and demonstrate significantly superior BLER performance at Eb/N0 = 2 dB across six [64,52] codes compared to existing block-wise approximation methods.

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 $Q_{ij}\neq0$. If $Q$ has half-bandwidth at most $ν$, this energy admits a trellis with at most $2^ν$ 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.

Problem

• GRAND
• maximum-likelihood decoding
• correlated Gaussian noise
• BPSK
• likelihood ordering

Innovation

• Low-Pathwidth GRAND
• maximum-likelihood decoding
• correlated Gaussian noise
• trellis enumeration
• quadratic pseudo-Boolean energy