[[378,12,27]] weight-7 coprime bivariate-bicycle code on Z_27 x Z_7

Direction & hypothesis

Objective: at 3 syndrome rounds under the board's depolarizing recipe at p = 0.002, interleaved 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-8 x unrestricted (no layout). This code comes from part 1, weight-6/7 two-block, 300 < n <= 420, k >= 12 of the third LER-objective search, the ground the first two searches (weight-6 two-block codes at n <= 300 with k >= 8; the existing board pool and generated weight-8 lifted-product, pair-partition, and non-abelian two-block codes at n <= 300) did not cover. The baseline's distance is set by the qubit budget (k12-d7: 12 copies of d = 7, 1164 qubits against the candidate's 756), and the decoder is the same BP+OSD configuration for both.

What was searched

Stage 1 admitted 579 candidates from part 1, weight-6/7 two-block, 300 < n <= 420, k >= 12, 707 candidates from part 2, coprime bivariate bicycle, weight 6/8, k >= 8, 24 candidates from part 3, hypergraph product, proven distance (1,310 generator hits with the required k, connected, and d >= 10 at 300 RIS trials; 0 repeats within the search, 0 already candidates of the first two searches, 0 exact board duplicates). Funnel: RIS at 2,000 trials (keep d >= 10; 271 of 271), RIS at 20,000 trials (247 kept), WL dedup against the board (20 dropped), 3-round circuit with the tier checks (257 built), GPU decode of 10,000 shots per basis at p = 0.002 against the matched surface baseline (6 of 11 beat it in both bases), the decoder-based d_circ estimate (6 kept, 0 more than 2 below d), then up to 100,000 shots per basis at p = 0.001 and 0.002 (fitted to a 90-minute GPU budget at the measured throughput) and a 300,000,000-trial ris_gpu pass per side.

Construction

Periodic bivariate-bicycle code on Z_27 x Z_7 (n = 2*l*m = 378): x = S_27 tensor I_7, y = I_27 tensor S_7 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^1y^2 + x^25y^1, B = x^0y^0 + x^12y^5 + x^17y^1 + x^19y^2; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(27, 7, [[0, 0], [1, 2], [25, 1]], [[0, 0], [12, 5], [17, 1], [19, 2]])). gcd(27, 7) = 1, so Z_27 x Z_7 is cyclic of order 189 and the code is the cyclic generalized-bicycle code over Z_189 with a(z) = z^0 + z^106 + z^163, b(z) = z^0 + z^12 + z^71 + z^100 (CRT relabeling x^a y^b -> z^t, t = a mod 27, t = b mod 7).

Schedule: interleaved; two-block interleaved (A: 3 terms, B: 4 terms), 8 CX layers per round; terms by bb_decompose translations on Z_27 x Z_7; X-check term slots [6, 1, 7, 2, 3, 5, 4], Z-check term slots [0, 6, 1, 3, 4, 2, 5] over terms A_0..A_2, B_0..B_3; RIS screen at 2 rounds: {'Z': 227, 'X': 246}

Evidence trail

RIS ladder for the submitted code (lightest logical found, both sides):

| RIS trials per side | seed | lightest logical found | |---|---|---| | 2,000 | 177059973 | 28 | | 20,000 | 770590826 | 27 | | 300,000,000 (GPU, verify/ris_gpu.py) | 1419984055 | X 27, Z 27 |

Claim: d <= 27, a witness-backed upper bound.

Decoder-based circuit fault-distance estimate at 3 rounds (a stacked BP+OSD search on the committed DEM, GPU, 2 seeds; our decoder-based estimator, not part of this repo): 58 (per basis {'Z': 58, 'X': 65}); an upper bound on d_circ.

Logical error rate per round at 3 rounds (exact 95 percent Poisson intervals on the failure count; the surface baseline is 12 copies of the distance-7 rotated surface code, 1164 physical qubits, geometric interleaved schedule, same noise recipe and decoder):

| p | basis | candidate | surface baseline | ratio | separated | |---|---|---|---|---|---| | 0.002 | Z | 7.69e-05 [3.52e-05, 1.46e-04] (9/39000) | 8.65e-04 [7.04e-04, 1.05e-03] (101/39000) | 0.089 | yes | | 0.002 | X | 2.56e-05 [5.29e-06, 7.49e-05] (3/39000) | 7.53e-04 [6.04e-04, 9.28e-04] (88/39000) | 0.034 | yes | | 0.001 | Z | 0 [0, 3.15e-05] (0/39000) | 2.56e-05 [5.29e-06, 7.49e-05] (3/39000) | 0.000 | no | | 0.001 | X | 0 [0, 3.15e-05] (0/39000) | 5.13e-05 [1.88e-05, 1.12e-04] (6/39000) | 0.000 | no |

Stage-4 screen (10,000 shots per basis at p = 0.002): Z 3 vs 22, X 1 vs 21 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-8 x unrestricted board; literature novelty UNVERIFIED. Duplicate check: exact None, WL None. Board cell ['weight-8', 'unrestricted']: board_advancing = True.

Dead ends

Of the candidates that reached the GPU screen, 5 did not beat their surface baseline in both bases at 10,000 shots; 0 had a d_circ estimate more than 2 below d; 0 produced no deterministic schedule or failed a tier check; 0 fell below d = 10 in the RIS ladder. Nearest stage-4 misses (candidate failures vs baseline failures, Z and X):

• [[420,8,49]] bivariate-bicycle (p2) vs k8-d9: Z 7 vs 4, X 10 vs 4
• [[420,8,48]] bivariate-bicycle (p2) vs k8-d9: Z 10 vs 4, X 5 vs 4
• [[406,8,44]] bivariate-bicycle (p2) vs k8-d9: Z 6 vs 4, X 14 vs 4
• [[400,16,8]] hypergraph-product (p3) vs k16-d5: Z 3593 vs 144, X 3617 vs 153
• [[400,16,8]] hypergraph-product (p3) vs k16-d5: Z 3751 vs 144, X 3672 vs 153

Tools

Claude (Claude Code, model Fable 5.1) as the agent in an unattended search: generation and circuits on the laptop, GPU decoding on RunPod A40 pods. Repo tooling: research/kit/bb.py, research/cyclic_gb.py, research/kit/group_algebra.py, and research/kit/products.py (constructions), research/kit/css.py and verify/gf2_fast.cpp (k and the RIS distance screen), verify/qldpc_verify.py fingerprint and WL signature (dedup against the board), the schedule chain of PR 1859's interleaved two-block builder (research/circuit_autogen.py), a SAT coloring, and the board's sequential builder with verify/circuit_verify.py's tier checks, our decoder-based d_circ estimator (not part of this repo), ler-pilot's rotated-surface matched baseline (pilot.py, build_interleaved.py), CUDA-Q QEC 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, verify/sat_certify.py for the exact check of part-3 distances, and verify/validate_candidate.py for the verdict.

Reproduction

import sys; sys.path.insert(0, "research/kit")
from bb import build_bb
HX, HZ = build_bb(l=27, m=7, A_terms=[[0, 0], [1, 2], [25, 1]], B_terms=[[0, 0], [12, 5], [17, 1], [19, 2]])   # [[378,12]]

Circuits: PR 1859's interleaved two-block builder (research/circuit_autogen.py) at 3 rounds, 8.0 CX layers per round, with the term slots listed under Construction. Decoding: strip the noise, reapply circuit_tools.apply_noise at p, derive the DEM, sample with the stim seed recorded in the receipts, decode with nv-qldpc-decoder as configured above; a failure is any logical observable decoded wrong.