A Finite-Blocklength Analysis for 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.

• The paper develops an ORB-RCU achievability bound that converts ORBGRAND’s rank-based decoding event into a tractable tail probability for general binary-input memoryless channels.
• The paper uses a Hoeffding decomposition and strong large-deviation analysis to show that rank-induced dependence yields a single-letter ORBGRAND dispersion and a third-order (ln n)/(2n) correction.
• The paper finds that, on BPSK-AWGN channels, the ORB normal approximation is accurate near n≈100 and incurs only a modest finite-blocklength penalty versus ML-RCU, such as n≤579 versus n≤545 at R=0.8C and ε=10⁻³ at 0 dB.

Overview

The paper develops a finite-blocklength information-theoretic analysis of ordered reliability bits Guessing Random Additive Noise Decoding (ORBGRAND) over general binary-input memoryless channels. Prior work established that ORBGRAND is nearly capacity-achieving for the BPSK-AWGN channel and exactly capacity-achieving via rank companding, but these results are asymptotic in blocklength and therefore silent in the short-to-moderate regime where GRAND-type decoders are most relevant, e.g., ultra-reliable low-latency communication (URLLC). The central technical obstacle is that ORBGRAND's decoding metric is rank-induced: it couples symbols through the ordering of LLR magnitudes and is not expressible as a sum of single-letter terms, so existing second-order analyses for additive (matched or mismatched) metrics do not apply.

The ORB-RCU bound

The authors first derive an ORBGRAND-specific random-coding union bound, termed ORB-RCU, on the ensemble-average error probability under i.i.d. uniform codebooks on $\{\pm1\}^n$. The bound mirrors the classical RCU construction but replaces the ML pairwise-error event by the event that a competing codeword's ORBGRAND metric does not exceed that of the transmitted codeword:

$$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$

Two structural observations make this tractable. First, because a competing codeword is independent of the channel output, its metric reduces to $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$ with i.i.d. Bernoulli($1/2$) variables $\mathsf{B}_i$; the conditional probability in the bound becomes the CDF $F_{\zeta_n}(n^2\mathsf{D})$ evaluated at the transmitted-codeword metric. Second, via the identity $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$ with $\mathsf{U}\sim\mathrm{Unif}[0,1]$, the entire bound collapses to a single tail probability involving $$\ln(M-1)+\ln F_{\zeta_n}(n^2\mathsf{D})-\ln\mathsf{U}$$.

Transmitted-codeword metric: U-statistics and Hoeffding decomposition

The transmitted-codeword metric $$\mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})$$ concentrates around $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$0, where $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$1 is the LLR magnitude, $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$2 its CDF, and $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$3 indicates a hard-decision error. A lemma establishes $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$4 under mild non-degeneracy conditions ($$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$5 and $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$6), using the probability integral transform to show $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$7 is uniform. Concentration of the metric within a fixed neighborhood of $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$8 holds with complement probability at most $$\mathrm{RCU}_{\mathrm{ORB}}(n,M)=\mathbb{E}\Big[\min\{1,(M-1)\Pr[\mathsf{D}(\underline{\hat{\mathsf{X}}},\underline{\mathsf{Y}})\le \mathsf{D}(\underline{\mathsf{X}},\underline{\mathsf{Y}})\mid \underline{\mathsf{X}},\underline{\mathsf{Y}}]\}\Big].$$9, via the Dvoretzky–Kiefer–Wolfowitz–Massart inequality for the empirical CDF plus Hoeffding's inequality.

The key representation treats $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$0 as an order-two U-statistic with asymmetric kernel $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$1. Applying the Hoeffding decomposition yields

$n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$2

with i.i.d., zero-mean $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$3 of variance $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$4, where $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$5 for an independent copy $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$6. The degenerate second-order term satisfies $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$7, so the ranking-induced dependence is absorbed into a negligible remainder. This decomposition is what licenses a Berry–Esseen argument despite the non-additivity of the metric.

Competing-codeword metric: strong large deviations

For the competing-codeword statistic $n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$8, the limiting cumulant generating function is obtained in closed form by a Riemann-sum argument,

$n^{-2}\sum_{i=1}^n i\,\mathsf{B}_i$9

and the left-tail probability $1/2$0 is converted to a right-tail event via sign flip. Invoking Joutard's strong large-deviation theorem—after verifying its assumptions uniformly over compact intervals $1/2$1, including analyticity on disks shifted to the saddlepoint and a lattice-span condition handled through a low-/high-frequency decomposition—the authors obtain, uniformly in $1/2$2,

$1/2$3

where $1/2$4 solves the saddlepoint equation $1/2$5 and $1/2$6 is the Legendre transform. This uniform prefactor-level expansion supplies the $1/2$7 third-order ingredient later visible in the rate expansion.

Second-order achievable rate and normal approximation

Combining the two metric characterizations with a Taylor expansion of $1/2$8 around $1/2$9 (the quadratic remainder is shown to satisfy $\mathsf{B}_i$0), a sandwich argument over the typical event, and a Berry–Esseen bound extended to random shifts, the paper arrives at the main result: for fixed $\mathsf{B}_i$1 and all sufficiently large $\mathsf{B}_i$2,

$\mathsf{B}_i$3

The first-order term $\mathsf{B}_i$4 is proved equal to the ORBGRAND generalized mutual information previously reported in asymptotic analyses, so the finite-blocklength theory is consistent with prior first-order results. The second-order coefficient defines the ORBGRAND dispersion,

$\mathsf{B}_i$5

which admits a single-letter variance representation despite the non-additive metric—a consequence of the Hoeffding projection. Truncating the expansion yields the ORB-normal approximation (ORB-NA),

$\mathsf{B}_i$6

A remark emphasizes that both $\mathsf{B}_i$7 and $\mathsf{B}_i$8 are decoder-induced quantities determined by the reliability-ordered guessing rule, not by the matched information density; the analysis is genuinely decoder-dependent rather than a specialization of mismatched-decoding results for additive metrics.

Numerical validation

For the BPSK-modulated AWGN channel, benchmarked against the meta-converse bound (saddlepoint approximation) and the ML-based RCU bound, three findings stand out. First, the ORB-RCU curve closely tracks the ML-RCU curve across rates at SNRs of 0 dB and 3 dB, indicating only a modest finite-blocklength penalty from rank-based ordering relative to ML decoding, with the gap shrinking as $\mathsf{B}_i$9 grows. Second, the ORB-NA tracks ORB-RCU accurately already at moderate blocklengths (around $F_{\zeta_n}(n^2\mathsf{D})$0). Third, minimal-blocklength computations quantify the loss concretely: at $F_{\zeta_n}(n^2\mathsf{D})$1 with $F_{\zeta_n}(n^2\mathsf{D})$2 and SNR 0 dB, ML-RCU certifies achievability for $F_{\zeta_n}(n^2\mathsf{D})$3 while ORB-RCU requires $F_{\zeta_n}(n^2\mathsf{D})$4; at $F_{\zeta_n}(n^2\mathsf{D})$5 with $F_{\zeta_n}(n^2\mathsf{D})$6, the corresponding figures are $F_{\zeta_n}(n^2\mathsf{D})$7 versus $F_{\zeta_n}(n^2\mathsf{D})$8. The dispersion curves show that $F_{\zeta_n}(n^2\mathsf{D})$9 closely tracks the ML dispersion $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$0 across the SNR range, both peaking near 0 dB; since $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$1 multiplies $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$2, the second-order backoff is most pronounced at moderate SNR and diminishes at high SNR. One caveat noted by the authors: occasional crossings where ORB-RCU dips slightly below ML-RCU do not contradict ML optimality, because RCU-type bounds involve different relaxations (notably the $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$3 clipping) and are not pointwise ordered.

Limitations and open questions

Several restrictions are acknowledged or implicit. The analysis covers the untruncated decoder ($\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$4); practical implementations truncate the query matrix, and the effect of a finite maximum number of queries on the finite-blocklength characterization is not addressed. The expansion requires $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$5 to lie in the interior of $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$6, excluding degenerate channels, and the remainder control is established for sufficiently large $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$7 even though numerics suggest accuracy already near $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$8. Only achievability is provided; no converse specific to ORBGRAND is derived, so the true decoder-constrained optimum could be tighter than ORB-RCU suggests. Finally, the paper leaves open the extension to guessing with explicit abandonment, following recent ensemble analyses for DMCs, which would characterize a finite-blocklength rate–reliability–complexity tradeoff including second-order refinements.

Conclusion

The paper supplies the first decoder-dependent finite-blocklength characterization of ORBGRAND, built from three technical components: an ORB-specific RCU bound, a Hoeffding-decomposition treatment of the rank-based transmitted-codeword metric, and a uniform strong large-deviation expansion for the competing-codeword metric. The resulting second-order expansion recovers the known ORBGRAND GMI at first order, introduces a single-letter dispersion $\mathbb{E}[\min\{1,A\}]=\Pr[A\ge \mathsf{U}]$9, and yields a normal approximation validated against meta-converse and ML-RCU benchmarks on the AWGN channel. The results quantify the modest finite-blocklength cost of rank-based noise guessing relative to ML decoding and provide a tractable surrogate for URLLC-oriented system evaluation without exhaustive simulation.