Reducing ORBGRAND Latency via Partial Gaussian Elimination

TL;DR

This work tackles tail-latency in ORBGRAND decoding by introducing elimination-aided ORBGRAND, which leverages the Rank of the Most Reliable Erroneous (RMRE) and partial Gaussian elimination (GE) to jointly verify EPs sharing the same RMRE. By performing column-wise GE and early consistency checks, the method filters out many infeasible EPs, reducing the total number of guesses and the GE complexity from $\mathcal{O}\!\left(N(N-K)^2\right)$ to $\mathcal{O}\left((N-K)M^2\right)$, where $M$ is the number of active columns. Simulation on BCH$(127,113)$ codes shows a reduction of 40–50% in average guesses with no loss in block error rate, enabling faster and more predictable decoding suitable for ultra-reliable low-latency communications. The approach is hardware-friendly, compatible with OSD for rare cases, and scalable for GRAND-based architectures in high-speed communication systems.

Abstract

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.

Figures (5)

• Figure 1: Comparison of the full and simplified GE schemes. (a1) - (a4): full elimination where each step updates all remaining columns. (b1) - (b4): proposed reduced elimination — (b1) processes only the first column together with $\underline{s}$. (b2) - (b3) apply the prior transformation $(I+Q^{(1)})$ and then eliminate the second column, and (b4) shows the resulting reduced GE, which operates only on the first three columns and $\underline{s}$, leaves all subsequent columns untouched.
• Figure 2: Comparison of BLER curves under different $E_b/N_0$s.
• Figure 3: Histogram of number of guesses for BCH(127, 113), $E_b/N_0$ = 5 dB.
• Figure 4: Distribution of $\text{RMRE}(\theta(\underline{Y}) \oplus \underline{W})$ and $M$.
• Figure 5: Comparison of average computational complexity under different $E_b/N_0$ values (1 floating-point operation = 8 XORs).

Theorems & Definitions (3)

• Definition 1
• Example 1
• Example 2