Performance Analysis and Optimal Design of ORB-Type GRAND Algorithms

2026-06-26

AI Summary

This work addresses ORB-type GRAND algorithms that rely solely on soft information for reliability ordering and proposes average guesswork posterior (AGP) as the optimal ordering criterion, recasting maximum-likelihood decoding as a ranking problem over the error pattern space within a unified analytical framework. For the first time, it decouples miss and overtaking errors based on AGP and derives exact block error rate (BLER) expressions for both random codes and fixed linear codes, revealing the impact of higher-weight codewords on performance. Leveraging these insights, the authors design an offline reordering scheme, RS-ORBGRAND, which significantly outperforms existing ORB-type GRAND algorithms; on the BCH(127,113) code, it operates within 0.1 dB of the maximum-likelihood bound at BLER = 10⁻⁶.

Abstract

Guessing Random Additive Noise Decoding (GRAND) performs decoding by sequentially guessing channel error patterns (EPs). Ordered Reliability Bits GRAND (ORBGRAND) is a notable instance suitable for efficient implementation, as it schedules EPs solely according to the ranking of soft channel outputs. In this paper, we generalize this principle to a broader class of GRAND algorithms whose testing order depends only on reliability ranking, referred to as ORB-type GRAND. We develop a unified analytical framework based on a key quantity termed the average guessing posterior (AGP), which captures the effectiveness of each EP and reduces decoding into an ordering problem over the EP space. For random code ensembles, we derive exact expressions for the block error rate (BLER), stopping-time distribution, and average number of tests under a fixed test budget. The analysis separates target-miss and target-preemption errors and shows that ordering EPs by non-increasing AGP is optimal over the EP set under consideration. For fixed linear block codes, we derive the BLER expression that isolates the code-dependent target-preemption term and characterize this term through higher-order weight relationships of codeword tuples, with a computable first-order upper bound as a useful special case. Guided by these insights, we formulate ReShuffled-ORBGRAND (RS-ORBGRAND) as an offline AGP-based reshuffling scheme. Numerical results for the Bose--Chaudhuri--Hocquenghem (BCH)$(127,113)$ code show that RS-ORBGRAND consistently improves existing ORB-type GRAND algorithms and lies within $0.1$~dB of a maximum-likelihood decoding lower-bound benchmark at a BLER of $10^{-6}$.

Problem

• GRAND
• ORBGRAND
• error pattern ordering
• block error rate
• decoding performance

Innovation

• GRAND
• ORBGRAND
• average guessing posterior
• block error rate
• error pattern ordering