A Finite-Blocklength Analysis for ORBGRAND

2026-03-08

AI Summary

This work addresses the lack of finite-blocklength performance analysis for ORBGRAND under short to moderate code lengths, a gap left by prior studies focusing solely on asymptotic regimes. We establish the first finite-blocklength theoretical framework for ORBGRAND, tackling the challenge posed by its non-additive, symbol-wise coupled ranking metric. Specifically, we introduce the ORB-RCU bound and derive single-letter expressions for generalized mutual information and channel dispersion. By characterizing the transmitted codeword metric via Hoeffding decomposition and analyzing competing codewords through large deviation theory combined with the Berry–Esseen theorem, we obtain a second-order achievable rate and a normal approximation. Numerical validation over BPSK-modulated AWGN channels demonstrates the tightness of the proposed bound and the accuracy of the approximation, confirming their relevance to practical communication scenarios.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Within the Guessing Random Additive Noise Decoding (GRAND) family, ordered reliability bits GRAND (ORBGRAND) has received considerable attention for its hardware-friendly exploitation of soft information. Existing information-theoretic results for ORBGRAND are asymptotic in blocklength and do not quantify its performance at short-to-moderate blocklengths. This paper develops a finite-blocklength analysis for ORBGRAND over general bit channel, addressing the key challenge that the rank-induced decoding metric is non-additive and coupled across symbols. We first derive an ORBGRAND-specific random-coding union (RCU)-type achievability (ORB-RCU) bound on the ensemble-average error probability. We then characterize two governing decoding metrics: the transmitted-codeword metric is treated as a U-statistic and analyzed via Hoeffding decomposition, while the competing-codeword metric is reduced to a weighted sum of independent and identically distributed Bernoulli random variables and analyzed through strong large-deviation analysis. Combining these ingredients with a Berry-Esseen argument yields a second-order achievable-rate expansion and the associated normal approximation, whose first-order term is shown to equal the ORBGRAND generalized mutual information and whose second-order term defines an ORBGRAND dispersion with a single-letter variance representation. Numerical results for BPSK-modulated additive white Gaussian noise channel validate the tightness of ORB-RCU relative to the maximum-likelihood based RCU benchmark and the accuracy of the normal approximation in the operating regime of practical interest.

Problem

• finite-blocklength
• ORBGRAND
• error probability
• non-additive metric
• short-to-moderate blocklengths

Innovation

• Finite-blocklength analysis
• ORBGRAND
• Random-coding union bound
• Second-order achievable rate
• Generalized mutual information