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A Finite-Blocklength Analysis for ORBGRAND

Zhuang Li, Wenyi Zhang

TL;DR

A finite-blocklength analysis for ORBGRAND over general bit channel is developed, addressing the key challenge that the rank-induced decoding metric is non-additive and coupled across symbols.

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.

A Finite-Blocklength Analysis for ORBGRAND

TL;DR

A finite-blocklength analysis for ORBGRAND over general bit channel is developed, addressing the key challenge that the rank-induced decoding metric is non-additive and coupled across symbols.

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.
Paper Structure (36 sections, 10 theorems, 176 equations, 5 figures, 1 table)

This paper contains 36 sections, 10 theorems, 176 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Channel Model
  4. ORBGRAND
  5. Random-Coding Union Bound and Decoding Metric Analysis
  6. Random-Coding Union Bound of ORBGRAND
  7. Transmitted-Codeword Metric Analysis
  8. Strong Large-Deviation Analysis for Competing-Codeword Metric
  9. Setup and Cumulant Generating Functions
  10. Right-Tail Strong Large-Deviations
  11. Left-Tail Strong Large-Deviations
  12. Second-Order Achievable Rate
  13. Taylor Expansion and Remainder Control
  14. Reduction to Unconditional Probability
  15. Gaussian Approximation with Random Perturbation
  16. ...and 21 more sections

Key Result

Theorem 1

For the random codebook generated i.i.d. according to $P_{\underline{\mathsf{X}}}$, the ensemble-average decoding error probability of ORBGRAND, denoted by $\epsilon(n,M)$, satisfies where and $(\underline{\mathsf{X}},\underline{\mathsf{Y}},\underline{\hat{\mathsf{X}}}) \sim P_{\underline{\mathsf{X}}}(\underline{x}) P_{\underline{\mathsf{Y}}|\underline{\mathsf{X}}}(\underline{y}| \underlin

Figures (5)

  • Figure 1: Average error probability bounds versus rate $R$ for the BPSK-modulated AWGN channel at $\mathrm{SNR}=0$ dB.
  • Figure 2: Average error probability bounds versus rate $R$ for the BPSK-modulated AWGN channel at $\mathrm{SNR}=3$ dB.
  • Figure 3: Rate-blocklength curves for the BPSK-modulated AWGN channel at $\mathrm{SNR}=0$ dB and $\epsilon=10^{-3}$.
  • Figure 4: Rate-blocklength curves for the BPSK-modulated AWGN channel at $\mathrm{SNR}=3$ dB and $\epsilon=10^{-6}$.
  • Figure 5: $V_{\mathrm{ORB}}$ and $V$ versus SNR for the BPSK-modulated AWGN channel.

Theorems & Definitions (13)

  • Theorem 1: ORB-RCU bound
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 2
  • Theorem 3
  • Theorem 4: Achievable lower bounds
  • Proposition 1
  • Remark 1: Difference from the classical normal approximation
  • Remark 2: Shifted-disk analyticity in Assumption (A.1)
  • ...and 3 more