A Finite-Blocklength Analysis for ORBGRAND

TL;DR

This paper provides a finite-blocklength analysis of ORBGRAND, a decoding method in the GRAND family, deriving bounds and approximations that quantify its performance at short-to-moderate blocklengths over general channels.

Contribution

It introduces a novel finite-blocklength analysis framework for ORBGRAND, including an RCU-type bound and a normal approximation, addressing the non-additive decoding metric challenge.

Findings

Numerical results show tightness of ORB-RCU bounds compared to maximum-likelihood benchmarks.

The normal approximation accurately predicts ORBGRAND performance in practical regimes.

The analysis reveals the mutual information and dispersion characteristics of ORBGRAND.

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.