AI 中文总结
针对有限记忆噪声效应后验，提出尾端校准软输出GRAND算法，实现后验权重等估计并给出相关理论界，可提供逐块APP估计等解码输出。
AI 中文摘要
在猜解随机加性噪声解码（GRAND）中，硬判决噪声效应的记忆会改变候选噪声效应的似然顺序。在软输出解码中，同样的记忆也会影响决定缺失列表概率的有限块量：当前列表之外的码本受限后验质量。现有关联感知GRAND方法利用局部依赖却未进行交织，但其停止规则和软输出规则并非源自有限记忆后验尾端。软输出GRAND（SOGRAND）为GRAND列表推导随机码本后验概率（APP）估计，但未提供用于关联噪声效应后验的后验权重、配分函数、尾端质量或逐位尾端边际的有限记忆算法。我们引入面向二元加性信道的尾端校准SOGRAND，其环境硬判决噪声效应后验在接收软信息条件下由有限记忆能量表示。解码器按后验能量非递减顺序枚举候选噪声效应，像GRAND一样查询码本成员资格，通过有限状态递归计算后验权重和尾端质量，并将未查询的码本受限分母估计为$p_qT_q$，其中$T_q$是环境后验尾端质量，$p_q$是剩余随机码本占用概率。在精确枚举且无弃权（不执行）的情况下，在定义后验能量的似然模型下，首个列出的码字为最大似然（ML）码字。我们还证明了环境后验尾端弃权（不执行）界，以及随机码本缺失列表估计器的条件无偏性、方差和集中界。相同的后验尾端分解为有限记忆噪声效应后验提供了逐块APP估计、缺失列表概率和逐位APP对数似然比（LLRs）。
英文摘要
In guessing random additive noise decoding (GRAND), memory in the hard-decision noise effect changes the likelihood order of candidate noise effects. In soft-output decoding, the same memory also affects the finite-block quantity determining the missing-list probability: the codebook-restricted posterior mass outside the current list. Existing correlation-aware GRAND methods exploit local dependence without interleaving, but their stopping and soft-output rules are not derived from finite-memory posterior tails. Soft-output GRAND (SOGRAND) derives random-codebook a posteriori probability (APP) estimates for GRAND lists, but does not provide finite-memory algorithms for posterior weights, partition functions, tail masses, or bitwise tail marginals for correlated noise-effect posteriors. We introduce Tail-Calibrated SOGRAND for binary additive channels whose ambient hard-decision noise-effect posterior, conditioned on received soft information, is represented by a finite-memory energy. The decoder enumerates candidate noise effects in nondecreasing posterior energy, queries codebook membership as in GRAND, computes posterior weights and tail masses by finite-state recursions, and estimates the unqueried codebook-restricted denominator as $p_qT_q$, where $T_q$ is the ambient posterior tail mass and $p_q$ is the remaining random-codebook occupancy probability. With exact enumeration and no abandonment, the first listed codeword is ML under the likelihood model defining the posterior energy. We also prove an ambient posterior-tail abandonment bound and, separately, conditional unbiasedness, variance, and concentration bounds for the random-codebook missing-list estimator. The same posterior-tail decomposition gives blockwise APP estimates, missing-list probabilities, and bitwise APP log-likelihood ratios (LLRs) for finite-memory noise-effect posteriors.
