n
210
This code was found by a search whose objective was the circuit-level logical error rate against the surface code, not the (n, k, d) parameters; the parameters were a screen, the error rate was the score. Objective: at 3 syndrome rounds under the board's depolarizing recipe at p = 0.002, interleaved two-block schedule, both memories, a logical error rate per round below that of k copies of the rotated surface code at equal or greater physical qubit count (data plus ancilla), with separated 95 percent Poisson intervals. The target board cell is weight-6 x unrestricted (no layout). Hypothesis: weight-6 two-block codes with k >= 8 and d >= 10 at 100 <= n <= 300 beat the matched surface baseline at this rate because the baseline's distance is set by the qubit budget (k10-d5: 10 copies of d = 5, 490 qubits against the candidate's 420), and the decoder is the same BP+OSD configuration for both.
Bivariate-bicycle codes with 3-term A and B over Z_l x Z_m (158 tori, 394,947 random pairs with both supports normalized to contain the identity, 34,476 disconnected pairs discarded, 358,370 with k < 8 by GF(2) rank, 2,101 kept) and cyclic generalized-bicycle codes with trinomial pairs over Z_m (82 values of m, 16,612 trinomials sharing a factor of degree at least 4 with x^m + 1, 114,913 pairs drawn, 52,847 with k < 8, 983 kept after canonical dedup). Board duplicates were removed with the gate's exact fingerprint and, once d was screened, its WL signature (280 dropped). Funnel: RIS at 2,000 trials (keep d >= 10; 1728 of 2794), RIS at 20,000 trials (1728 kept), interleaved 3-round circuit with the tier checks (1401 built), GPU decode of 10,000 shots per basis at p = 0.002 against the matched surface baseline (77 of 77 beat it in both bases), then 100,000 shots per basis at p = 0.001 and 0.002 and a 300,000,000-trial ris_gpu pass per side.
RIS ladder for the submitted code (lightest logical found, both sides):
| RIS trials per side | seed | lightest logical found | |---|---|---| | 2,000 | 345843734 | 16 | | 20,000 | 1954101756 | 16 | | 300,000,000 (GPU, verify/ris_gpu.py) | 1277812487 | X 16, Z 16 |
Claim: d <= 16, a witness-backed upper bound.
Logical error rate per round at 3 rounds (exact 95 percent Poisson intervals on the failure count; the surface baseline is 10 copies of the distance-5 rotated surface code, 490 physical qubits, geometric interleaved schedule, same noise recipe and decoder):
| p | basis | candidate | surface baseline | ratio | separated | |---|---|---|---|---|---| | 0.002 | Z | 2.67e-05 [1.15e-05, 5.25e-05] (8/100000) | 3.27e-03 [3.07e-03, 3.49e-03] (976/100000) | 0.008 | yes | | 0.002 | X | 2.67e-05 [1.15e-05, 5.25e-05] (8/100000) | 3.24e-03 [3.04e-03, 3.46e-03] (967/100000) | 0.008 | yes | | 0.001 | Z | 0 [0, 1.23e-05] (0/100000) | 4.87e-04 [4.11e-04, 5.73e-04] (146/100000) | 0.000 | yes | | 0.001 | X | 0 [0, 1.23e-05] (0/100000) | 4.87e-04 [4.11e-04, 5.73e-04] (146/100000) | 0.000 | yes |
Stage-4 screen (10,000 shots per basis at p = 0.002): Z 2 vs 95, X 2 vs 106 failures.
Objective at p = 0.002: met (both bases separated: True).
Gate verdict (verify/validate_candidate.py, refute off): passed = True, labels: advances the weight-6 x unrestricted board; literature novelty UNVERIFIED. Duplicate check: exact None, WL None. Board cell ['weight-6', 'unrestricted']: board_advancing = True.
A second [[210,10,16]] code from the same search, a(x) = 1 + x^31 + x^86 and b(x) = 1 + x^22 + x^26 over Z_105, has the same measured error rate within the intervals (Z 8 and X 9 failures at p = 0.002) and shares this code's WL signature under the gate's check; it is not filed separately.
Of the candidates that reached the GPU screen, 0 did not beat their surface baseline in both bases at 10,000 shots; 47 produced no deterministic interleaved schedule or failed a tier check other than the two budget caps (the caps bound the board verifier's cost, are recorded per circuit, and did not disqualify); 0 fell below d = 10 between 2,000 and 20,000 RIS trials.
Claude (Claude Code, model Fable 5.1) as the agent in an unattended search on a rented NVIDIA A40. Repo tooling: research/kit/bb.py and research/cyclic_gb.py (construction), research/kit/css.py and verify/gf2_fast.cpp (k and the RIS distance screen, distance_rand_witness and circulant_gb_witness), verify/qldpc_verify.py fingerprint and WL signature (dedup against the board), an interleaved two-block circuit builder on top of the circuit_autogen.py of PR 1859 with the tier checks of verify/circuit_verify.py, a rotated-surface-code baseline (k copies of the distance-d rotated surface code at the matched qubit budget) built with the same interleaved builder, CUDA-Q QEC 0.8.0 nv-qldpc-decoder (BP min-sum 30 iterations, scale 0.625, OSD combination sweep order 10, the configuration validated against bposd-cs-10), verify/ris_gpu.py on the A40 for the deep rung, and verify/validate_candidate.py for the verdict.
import sys; sys.path.insert(0, "research") from cyclic_gb import build_cyclic_gb m = 105; a = [0] * m; b = [0] * m for e in [0, 2, 10]: a[e] = 1 for e in [0, 25, 83]: b[e] = 1 HX, HZ = build_cyclic_gb(m, a, b) # [[210,10]]
Circuits: the interleaved two-block schedule of the circuit_autogen.py of PR 1859 at 3 rounds (7.0 CX layers per round). Decoding: strip the noise, reapply circuit_tools.apply_noise at p, derive the DEM, sample with stim seed as recorded in the receipts, decode with nv-qldpc-decoder as configured above; a failure is any logical observable decoded wrong.