A Finite-Blocklength Analysis for ORBGRAND
This work is significant for researchers and engineers working on error-correction codes, providing a more accurate performance quantification for ORBGRAND at practical short-to-moderate blocklengths, which was previously only available asymptotically.
This paper provides a finite-blocklength analysis for ORBGRAND, a hardware-friendly decoding method, addressing the challenge of its non-additive decoding metric. It derives an ORBGRAND-specific random-coding union (ORB-RCU) bound and a second-order achievable-rate expansion, showing its tightness against maximum-likelihood benchmarks and accuracy of normal approximation.
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.