Reducing ORBGRAND Latency via Partial Gaussian Elimination

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.

💡 Research Summary

The paper addresses a critical latency bottleneck in Ordered Reliability Bits GRAND (ORBGRAND), a variant of the universal Guessing Random Additive Noise Decoding (GRAND) framework. While ORBGRAND leverages only the ordering of log‑likelihood ratio (LLR) magnitudes to generate error‑pattern (EP) queries, its worst‑case latency can explode under adverse channel conditions because the number of candidate EPs that must be examined grows dramatically. To mitigate this, the authors introduce two complementary ideas: (1) the Rank of the Most Reliable Erroneous (RMRE) bit, and (2) a partial Gaussian elimination (GE) filtering stage.

RMRE is defined as the index (in the LLR‑sorted order) of the most reliable bit that is flipped in a given EP. Empirically, EPs with smaller RMRE values dominate the early part of the search, because the most reliable bits are less likely to be erroneous. By grouping all EPs that share the same RMRE, the decoder can test an entire group in a single step rather than probing each EP individually. This grouping alone can eliminate a large fraction of useless guesses.

The second contribution is a partial GE procedure that operates on the parity‑check matrix H after it has been reordered according to the same LLR ranking used by ORBGRAND. Traditional full GE would require O(N(N‑K)²) operations, where N is the code length and K the information length. The authors instead perform column‑wise elimination only on the first k columns, where k is the smallest number of columns needed to resolve the syndrome for a given RMRE. By stopping the elimination once a feasible RMRE is found, the algorithm avoids unnecessary work on the remaining columns. The resulting complexity is reduced to O((N‑K)·M²), with M typically 20–30 % of N‑K.

Algorithmically, the decoder proceeds as follows:

1. Compute the syndrome s = H·θ(y) from the hard‑decision vector θ(y).
2. Sort the received LLRs, obtain the permutation matrix P, and form the permuted parity‑check matrix Hπ = H·P.
3. Perform partial GE column by column. After processing the n‑th column, check whether the rank of the sub‑matrix formed by the first n columns equals the rank of the same sub‑matrix augmented with the syndrome. If equality holds, a solution exists that flips only the first n bits; this corresponds to EPs with RMRE = n.
4. Enumerate all such EPs (denoted En) and intersect them with the pre‑stored ORBGRAND candidate list. The smallest index t* among the intersecting EPs is selected as the first valid guess.
5. Resume the ordinary ORBGRAND search, but skip all EPs whose RMRE ≤ n, because they have already been ruled out by the GE filter.

The authors provide a detailed complexity analysis. Let T₁ be the average number of guesses required by vanilla ORBGRAND, and T₂ the number after filtering. The verification cost per guess is c·N, where c is the average number of parity‑check equations evaluated. The partial GE stage incurs an additional cost proportional to the expected number of pivots M: E