Elimination-Aided ORBGRAND Scheme

• The paper introduces a novel elimination-aided ORBGRAND framework that integrates RMRE statistics with partial Gaussian elimination to filter error patterns efficiently.
• It demonstrates up to 50% reduction in tail latency and a 40–55% decrease in syndrome checks while maintaining maximum-likelihood error correction performance.
• The methodology enables hardware-friendly, parallel decoding through group skipping and incremental Gaussian elimination, making it ideal for URLLC applications.

The elimination-aided ORBGRAND scheme is a low-complexity, hardware-friendly decoding framework designed to enhance the latency and efficiency of universal block-code decoders. Building on the Ordered Reliability Bits Guessing Random Additive Noise Decoding (ORBGRAND) algorithm, this method introduces a mechanism for filtering groups of error patterns (EPs) through the integration of the Rank of the Most Reliable Erroneous (RMRE) bit and partial Gaussian-elimination filtering. This architecture significantly reduces both the average and the worst-case decoding latency while preserving maximum-likelihood error correction performance, making it particularly suitable for ultra-reliable low-latency communication (URLLC) scenarios (Wan et al., 1 Feb 2026).

1. Foundations of ORBGRAND and Latency Challenges

ORBGRAND extends the universal Guessing Random Additive Noise Decoding (GRAND) paradigm by leveraging soft detection information. In a binary linear block code of length $N$ and dimension $K$, received sequences are first processed into log-likelihood ratios (LLRs):

$$L_i = \log \frac{P(y_i \mid x_i=0)}{P(y_i \mid x_i=1)}, \quad i=1,\dots,N$$

After a hard decision $\theta(y_i)$, ORBGRAND generates a fixed, precomputed list of EPs. The absolute LLRs are sorted, and the list is permuted according to the reliability ordering. Decoding proceeds by testing these candidate EPs in order, stopping at the first valid codeword (i.e., when the syndrome matches):

$$H_\pi\,\tilde e(t) = s, \quad H_\pi = H\mathcal{P}, \quad s = H\theta(y)$$

Though average decoding is fast at moderate/high SNR, tail latency can become problematic under unfavorable channel conditions due to long worst-case EP search depths, which is detrimental in latency-limited applications (Wan et al., 1 Feb 2026).

2. Rank of the Most Reliable Erroneous Bit (RMRE) Statistic

RMRE quantifies, for a given EP $e \in \{0,1\}^N$, the largest index (w.r.t. increasing reliability) where a bit flip occurs:

$\mathrm{RMRE}(e) = \max\{ i : e_{r_i} = 1 \}$

where $r_1$ is the least reliable and $r_N$ the most reliable bit per the sorted LLRs. For the true error vector $\hat e = \theta(y) \oplus \hat w$, RMRE identifies the position of the most reliable bit in error. Patterns with small RMRE are concentrated among less reliable bits and have higher posterior likelihood (Wan et al., 1 Feb 2026).

3. Partial Gaussian-Elimination Filtering

The elimination-aided ORBGRAND scheme employs a column-permutation $\mathcal{P}$ on the code’s parity-check matrix $H$, yielding:

$H_\pi = H\mathcal{P} = [h_{r_1} \ \cdots\ h_{r_N}]$

Partial Gaussian elimination (GE) is performed column-wise, only as far as necessary to decide the consistency of syndromes for solutions up to rank $n$ (i.e., using the first $n$ columns). After each pivot is established at column $n$, the process checks if the reduced system admits a solution supported on the first $n$ bits via the rank equality:

$$\mathrm{rank}\bigl([\;h^{(k)}_1,\dots,h^{(k)}_n\;]\bigr) = \mathrm{rank}\bigl([\;h^{(k)}_1,\dots,h^{(k)}_n,\;s^{(k)}\;]\bigr)$$

If satisfied, all EPs with support within the first $n$ bits and RMRE $= n$ are jointly verified. If none are codewords, all pre-stored EPs of RMRE $\leq n$ are eliminated from further consideration (Wan et al., 1 Feb 2026).

4. Decoding Workflow and Joint Filtering

The overall decoding strategy incorporates the following steps:

• Compute and sort LLRs, deriving the permutation $\mathcal{P}$ and $H_\pi$.
• For each ascending value of $n$, use partial GE to determine the feasibility of solutions with RMRE $= n$.
• Extract the set $\mathcal{E}_n$ of valid EPs of that class and intersect with the ORBGRAND EP table.
• If a match is found, decode; otherwise, bulk-eliminate all candidates with RMRE $\leq n$.

This joint filtering enables the skipping of large groups of low-likelihood EPs in a single (incremental) GE operation, in contrast to the serial, one-by-one syndrome checking of conventional ORBGRAND. The incremental nature of column-wise GE confines computational cost, with the dominant term governed by the number of pivots, $M$, needed for the actual error pattern; see the asymptotic cost expressions below (Wan et al., 1 Feb 2026):

$$\mathcal{O}_{\mathrm{elim}} = \frac{1}{2}M(M+3)(N-K) + T_2\,c\,N$$

with $T_2$ being the number of EP checks after filtering ($T_2 \leq T_1$ from standard ORBGRAND).

5. Complexity, Latency, and Empirical Results

Simulation studies reveal that elimination-aided ORBGRAND filters more than 50% of EPs, yielding 40–55% reductions in the number of syndrome checks and up to 50% decreases in tail latency, with no observed loss in block error rate (BLER) compared to original ORBGRAND (Wan et al., 1 Feb 2026). Performance metrics from simulations with BCH(127,113) under AWGN/BPSK (at $E_b/N_0 = 4,5,6$ dB) exhibit:

• Identical BLER curves to baseline ORBGRAND down to $10^{-6}$
• At 5 dB, an average guess count reduction from $9.67 \times 10^1$ to $4.49 \times 10^1$ (53.5% decrease)
• Floating-point operation (measured via XORs) reduction exceeding 45% at 5 dB
• The RMRE statistic is strongly skewed toward small values under moderate SNR, ensuring the bulk of GE effort is localized to early (low-rank) bits, with negligible impact on total complexity

This demonstrates suitability for URLLC use cases demanding tight latency constraints without BLER compromise.

6. Broader Context and Hardware Considerations

Elimination-aided ORBGRAND is compatible with high-throughput, parallel decoding architectures. Its filtering stage leverages only XOR logic and incremental GE, enabling straightforward mapping to parallel hardware. The method’s bit-level parallelism synergizes with other tree-based and pruning acceleration techniques used in advanced GRAND and hybrid decoders (Wan et al., 2 Oct 2025).

Integration with tree-based EP representations and parallel exploration/pruning, as in the hybrid approaches, further strengthens its appeal in scalable, ML-optimal architectural regimes. Notably, the elimination-aided mechanism stands out by providing group-skipping without disturbing the ORBGRAND ordering principle, maintaining both maximum-likelihood accuracy and implementation simplicity.

7. Relationships to Prior Work and Limitations

Unlike SGRAND and other sequential ML decoders, ORBGRAND employs only the rank vector of LLRs, yielding a significant reduction in soft-information bandwidth requirements while remaining agnostic to the magnitude distribution (Duffy, 2020). The elimination-aided scheme overlays a group-filtering layer atop ORBGRAND, removing entire classes of error patterns by leveraging parity-check structure.

A plausible implication is that, in very high-reliability or low-complexity deployments, the partial GE pruning may be adjusted or omitted if tail latency is not a primary constraint. However, care is required to reconcile elimination-aided filtering with other dynamic tree-based or partition-based enumeration algorithms, so as not to violate the underlying ordering required for soft-ML optimality (Duffy, 2020).

In summary, elimination-aided ORBGRAND achieves hardware-efficient, low-latency universal decoding through the fusion of RMRE-based bit-reliability statistics and incremental algebraic verification, substantiated by extensive empirical and theoretical analysis for modern block codes (Wan et al., 1 Feb 2026).