Soft GRAND under Channel Uncertainty: Minimax Posterior-Envelope Ordering and Gallager-Exponent Preservation
Abstract
Soft Guessing Random Additive Noise Decoding queries corrections in an observation-dependent order. For a known channel, the matched order ranks corrections by nonincreasing conditional posterior probability. Under channel uncertainty, admissible channels may induce matched orders that assign different ranks to the same correction at the same observation. We instead use a posterior-envelope order, defined by the pointwise supremum of codebook-independent uniform-input correction posteriors. For fixed measurable posterior representatives and a measurable envelope, this order minimizes, at each observation, the worst-case logarithm of the rank--posterior product over admissible channel--correction pairs with positive posterior probability. For uniform-subset codebooks, this finite-block minimax property does not by itself determine the ensemble-average error exponent, which depends on the distribution of the realized correction rank. For i.i.d. correction--observation pairs with a finite correction alphabet whose joint distribution is dominated by the product of counting measure on that alphabet and a σ-finite measure on the observation space, Arimoto conditional Rényi entropy characterizes the matched rank-moment spectrum. An auxiliary posterior-envelope order preserves this spectrum on [0,1] when its posterior family contains the corresponding conditional power tilts and has subexponential conditional Shtarkov complexity on observation sets whose complement probabilities decay exponentially faster than the reciprocal correction-space cardinality. Under additional differentiability and entropy nondegeneracy conditions, the auxiliary order attains the matched uniform-subset ensemble-average error exponent at every fixed normalized code rate strictly between zero and one. For uniform-input memoryless channels, this exponent equals Gallager's uniform-input random-coding exponent.
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BibTeX
@misc{razeghi2026soft,
title = {{Soft GRAND under Channel Uncertainty: Minimax Posterior-Envelope Ordering and Gallager-Exponent Preservation}},
author = {Behrooz Razeghi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29688},
eprint = {2609.29688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29688},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Soft GRAND under Channel Uncertainty: Minimax Posterior-Envelope Ordering and Gallager-Exponent Preservation
AU - Behrooz Razeghi
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.29688
UR - https://arxiv.org/abs/2609.29688
AB - Soft Guessing Random Additive Noise Decoding queries corrections in an observation-dependent order. For a known channel, the matched order ranks corrections by nonincreasing conditional posterior probability. Under channel uncertainty, admissible channels may induce matched orders that assign different ranks to the same correction at the same observation. We instead use a posterior-envelope order, defined by the pointwise supremum of codebook-independent uniform-input correction posteriors. For fixed measurable posterior representatives and a measurable envelope, this order minimizes, at each observation, the worst-case logarithm of the rank--posterior product over admissible channel--correction pairs with positive posterior probability. For uniform-subset codebooks, this finite-block minimax property does not by itself determine the ensemble-average error exponent, which depends on the distribution of the realized correction rank. For i.i.d. correction--observation pairs with a finite correction alphabet whose joint distribution is dominated by the product of counting measure on that alphabet and a σ-finite measure on the observation space, Arimoto conditional Rényi entropy characterizes the matched rank-moment spectrum. An auxiliary posterior-envelope order preserves this spectrum on [0,1] when its posterior family contains the corresponding conditional power tilts and has subexponential conditional Shtarkov complexity on observation sets whose complement probabilities decay exponentially faster than the reciprocal correction-space cardinality. Under additional differentiability and entropy nondegeneracy conditions, the auxiliary order attains the matched uniform-subset ensemble-average error exponent at every fixed normalized code rate strictly between zero and one. For uniform-input memoryless channels, this exponent equals Gallager's uniform-input random-coding exponent.
ER -