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arXiv:2606.28030

Performance Analysis and Optimal Design of ORB-Type GRAND Algorithms

Li Wan, Wenyi Zhang

Published Jun 26, 2026• 0 citations
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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 $10^{-6}$.

DOI
10.48550/arxiv.2606.28030

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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)(127,113)(127,113) code show that RS-ORBGRAND consistently improves existing ORB-type GRAND algorithms and lies within 0.10.10.1~dB of a maximum-likelihood decoding lower-bound benchmark at a BLER of 10−610^{-6}10−6.
Guessing Random Additive Noise Decoding (GRAND) is a universal framework for decoding all block codes by testing candidate error patterns (EPs). Ordered Reliability Bits GRAND (ORBGRAND) facilitates parallel implementation of GRAND by exploiting log-likelihood ratio (LLR) rankings but still suffers from high tail latency under unfavorable channel conditions, limiting its use in real-time systems. We propose an elimination-aided ORBGRAND scheme that reduces decoding latency by integrating the Rank of the Most Reliable Erroneous (RMRE) bit with a partial Gaussian-elimination (GE) filtering mechanism. The scheme groups and jointly verifies EPs that share the same RMRE, and once a valid EP is identified, the ORBGRAND search is resumed. By leveraging prior GE steps to filter out unnecessary guesses, this approach significantly reduces the number of EPs to be tested, thereby lowering both average and worst-case latency while maintaining error-correction performance. Simulation results show that compared to the original ORBGRAND, the elimination-aided ORBGRAND filters out more than 50\% of EPs and correspondingly reduce overall computational complexity, all with no loss in block error rate. This demonstrates that this approach is suitable for ultra-reliable low-latency communication scenarios.
Guessing Random Additive Noise Decoding (GRAND) is a universal decoding algorithm that has been recently proposed as a practical way to perform maximum likelihood decoding. It generates a sequence of possible error patterns and applies them to the received vector, checking if the result is a valid codeword. Ordered reliability bits GRAND (ORBGRAND) improves on GRAND by considering soft information received from the channel. Both GRAND and ORBGRAND have been implemented in hardware, focusing on average performance, sacrificing worst case throughput and latency. In this work, an improved pattern schedule for ORBGRAND is proposed. It provides>0.5> 0.5>0.5dB gain over the standard schedule at a block error rate≤10−5\le 10^{-5}≤10−5, and outperforms more complex GRAND flavors with a fraction of the complexity. The proposed schedule is used within a novel code-agnositic decoder architecture: the decoder guarantees fixed high throughput and low latency, making it attractive for latency-constrained applications. It outperforms the worst-case performance of decoders by orders of magnitude, and outperforms many best-case figures. Decoding a code of length 128, it achieves a throughput of 79.21 Gb/s with 58.49 ns latency, yielding better energy efficiency and comparable area efficiency with respect to the state of the art.
Guessing random additive noise decoding (GRAND) is a universal decoding paradigm that decodes by repeatedly testing error patterns until identifying a codeword, where the ordering of tests is generated by the received channel values. On one hand, while testing error patterns in a descending order of posterior probabilities leads to maximum likelihood decoding, its implementation complexity is prohibitive. On the other hand, testing error patterns with a prescribed set of error patterns permuted by the ranking among magnitudes of log-likelihood ratios (i.e., ordered reliability bits, ORB) enables efficient implementation, but results in performance loss for finite-length codes. Aiming at harnessing the strengths of these two approaches, this work proposes a fine-tuning method to improve ORBGRAND, adjusting the ordering of tests with the aid of very few exact channel soft values. This method is based on a metric for assessing the "well-orderedness" of error patterns. The metric is studied via the lens of the asymptotic theory of integer partitioning, which provides highly accurate estimation in numerical experiments. The metric then leads to an effective identification of fine-tuning to conduct, at the cost of a negligible increment of complexity. Numerical experiments demonstrate that the proposed fine-tuning method achieves a substantial performance enhancement compared with ORBGRAND.
Parallelism has become a central concern in modern decoding frameworks aiming to meet stringent throughput and latency requirements. Guessing Random Additive Noise Decoding (GRAND) is a recently proposed decoding paradigm that tests candidate Error Patterns (EPs) until a valid codeword is found. Among its variants, Soft GRAND (SGRAND) achieves maximum-likelihood (ML) decoding but relies on real-time generation and likelihood ordering of EPs, making parallel execution nontrivial under the ML optimality constraint. In this work, we introduce a unified binary tree representation of EPs, termed the EP tree, which formalizes the hierarchical structure underlying SGRAND and Ordered Reliability Bits (ORB) GRAND algorithms, enabling structured organization of EPs and algorithmic-level parallel exploration. Building upon this unified framework, we propose a parallel design of SGRAND that preserves ML optimality while significantly reducing decoding complexity through pruning strategies and tree-based computation. Furthermore, we develop an enhanced ORBGRAND algorithm based on the same EP tree representation, improving decoding performance toward ML while retaining parallel efficiency. Numerical experiments show that the proposed parallel SGRAND achieves a 3.96× reduction in decoding latency compared with its serial counterpart, while the enhanced ORBGRAND achieves a 4.21× speedup, demonstrating the effectiveness of the unified tree-based framework and its strong potential for future algorithmic and hardware optimizations.
Guessing random additive noise decoding (GRAND) is a recently proposed decoding approach, and ordered reliability bits GRAND (ORBGRAND) is one of its variants, exploiting channel soft information by sorting error patterns based upon the rank statistics of reliability values of received samples. In this paper, the ORB heuristic is employed to improve belief propagation (BP) decoding of polar codes. Specifically, the improved logistic weight ordering (iLWO) rule of ORB is employed to locate unreliable received samples, and these unreliable samples are flipped and scaled, followed by standard BP decoding. The performance gain of such ORB-enhanced BP decoding can be as high as 1.7dB at block error rate of 10–4with an average number of queries of 1.5.
Error correction techniques traditionally focus on the co-design of restricted code-structures in tandem with code-specific decoders that are computationally efficient when decoding long codes in hardware. Modern applications are, however, driving demand for ultra-reliable low-latency communications (URLLC), rekindling interest in the performance of shorter, higher-rate error correcting codes, and raising the possibility of revisiting universal, code-agnostic decoders. To that end, here we introduce a soft-detection variant of Guessing Random Additive Noise Decoding (GRAND) called Ordered Reliability Bits GRAND that can accurately decode any moderate redundancy block-code. It is designed with efficient circuit implementation in mind, and determines accurate decodings while retaining the original hard detection GRAND algorithm's suitability for a highly parallelized implementation in hardware. ORBGRAND is shown to provide excellent soft decision block error performance for codes of distinct classes (BCH, CA-Polar and RLC) with modest complexity, while providing better block error rate performance than CA-SCL, a state of the art soft detection CA-Polar decoder. ORBGRAND offers the possibility of an accurate, energy efficient soft detection decoder suitable for delivering URLLC in a single hardware realization.