Choosing a decoder

Three paradigms

All three start from belief propagation (BP), which alone often fails to converge on degenerate qLDPC codes. What follows BP is the differentiator:

• BP + OSD-CS — repairs a non-converged BP result by information-set inversion (ordered-statistics decoding with a combination sweep).
• relay-BP — drops post-processing entirely; it uses disordered memory plus relaying and ensembling to make BP itself converge, in a real-time iteration budget.
• BP + OTF — an inversion-free post-processor that builds a spanning forest (Ordered Tanner Forest) on which BP is exact, in almost-linear time.

At a glance

BP + OSD-CS

The validated baseline and Glean's default. It is the most accurate general-purpose choice and the right starting point for any new DEM. Its cost is the inversion step (O(n³) in the worst case), which makes it an offline / analysis decoder rather than a real-time one. You already used it in the previous guide via decode_batch(..., osd_order=10).

✓ B3 — == Roffe's ldpc to 99.8% / <0.4σ on the gross code.

relay-BP

The OSD-free, real-time differentiator. Reach for it when you need a bounded per-shot latency (decoding has to keep up with ~µs QEC cycles) or when ensembling buys you accuracy. It is a drop-in swap on the same DEM — call decode_batch_relay instead:

# Same DEM 'g' and syndromes 'det' as the previous guide — just a different call.
pred, iters = g.decode_batch_relay(
    det.tolist(), gamma0=0.1, pre_iter=80, num_sets=60, set_max_iter=60,
    gamma_dist=(-0.24, 0.66), stop_nconv=5, alpha=1.0, seed=0, parallel=True)

# 'iters[i]' is the total BP iterations spent on shot i — the real-time budget metric.
import numpy as np
print("median iters", int(np.median(iters)), " p90", int(np.percentile(iters, 90)))

The headline real-time metric is the per-shot BP iteration count returned alongside the predictions — not wall-clock µs. stop_nconv is the "-S" in "Relay-BP-S": the number of converged relay legs to ensemble (higher = more accurate, more work in the rare tail). num_sets is the number of relay legs.

✓ B4 — == the authors' relay_bp crate (<0.95σ); median 16 / p90 212 iters ≪ 600 budget.

BP + OTF

The inversion-free third paradigm. Its appeal is the almost-linear O(n log n) post-processor, an asymptotic latency edge that should show up at large n. It needs an offline-built sparsified DEM plus a transfer-matrix incidence, produced by benchmarks/otf_sparsify.py; build the decoder with glean.load_bp_otf(...). A single OTF decoder undershoots BP+OSD-0 by design — the published accuracy is recovered by a 23-member ensemble (see benchmarks/bp_otf_gross.py for the end-to-end build).

✓ B7 — ensemble BP+OTF == Glean's BP+OSD-0, 0.33σ.

How to pick

• Just want a correct, accurate answer? Start with BP + OSD-CS-10. It is the default for a reason.
• Decoding in real time, or on hardware? Use relay-BP and watch the per-shot iteration budget rather than wall-clock.
• Pushing to very large codes where inversion dominates? Evaluate BP + OTF; its scaling advantage grows with n.

Whichever you pick, the deepest logical error rates are out of reach for direct Monte-Carlo. The next guide shows how Glean's estimators get there — and they work with any of these three decoders unchanged.