AI 中文总结
针对相关高斯噪声下BPSK传输，提出低路径宽度GRAND（LP-GRAND），其按非递增能量枚举模式，经实验验证，在10000帧中与穷举ML结果一致，且BLER优于基于块的近似。
AI 中文摘要
软输入GRAND的有限块最大似然（ML）保证需要按非递增条件似然顺序查询噪声效应模式。在相关高斯噪声下，加性可靠性度量和独立块近似无法保持该顺序，因为匹配度量包含跨坐标交互；首次码本命中不一定产生ML码字。我们开发了针对精度矩阵Q的二进制相移键控（BPSK）的低路径宽度GRAND（LP-GRAND）。高斯负对数似然的候选相关部分是依赖观测的二次伪布尔能量，其交互图在Q_ij≠0时恰好有边{i,j}。若Q的半带宽最多为ν，该能量在每层最多有2^ν个状态的网格；宽度为w的路径分解在每层产生最多2^{w+1}个包分配。在实数运算中，后缀动态规划和最佳优先完整路径枚举按非递增能量枚举模式。通过完整枚举且无弃权（不执行），对于任意非空等概率码字的二进制码本，首次码本命中会产生ML码字。对于两个[20,12]码，LP-GRAND在所有10000帧中都与穷举码字ML结果一致；在标称E_b/N_0=2dB下，其针对六个[64,52]码的经验BLER低于每个基于块的近似的BLER。
英文摘要
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.
