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

2026-08-04

AI Summary

This work addresses the limitations of existing soft-output GRAND methods in modeling the posterior tail and establishing effective stopping criteria under finite-memory correlated noise, which hinder accurate estimation of missing list probabilities and bit-level posteriors. Focusing on binary additive channels, the paper proposes the first soft-output GRAND framework with tail calibration: it employs a finite-memory energy-based model to characterize the noise posterior, enumerates candidate noise sequences in order of increasing energy, and combines codebook queries with finite-state recursion to efficiently compute posterior weights. A stochastic codebook placeholder model is introduced to estimate the normalization constant for unqueried candidates. The method yields exact block- and bit-level posteriors, log-likelihood ratios (LLRs), and missing list probabilities; without any abandonment strategy, the first decoded codeword is guaranteed to be maximum-likelihood. Theoretical guarantees are provided for tail truncation error bounds and the unbiasedness, variance, and concentration of the estimators.

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.

Problem

• finite-memory noise
• soft-output decoding
• posterior tail
• GRAND
• correlated noise

Innovation

• Tail-Calibrated SOGRAND
• finite-memory posterior
• soft-output decoding
• posterior tail mass
• random-codebook APP estimation