Tail-Calibrated Soft-Output GRAND for Finite-Memory Noise-Effect Posteriors

Abstract

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.