n
216
Paper: arXiv:2610.06623, "Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes" (headline non-CSS PBB result; match W5PBB-216-4-10-36c5ce3b).
Target: the general-stabilizer board's unrestricted x weight-6 (and looser) cells, one below the ceiling at max check weight 5. Same PBB construction as the companion [[144,12,12]] submission (block-1 z-part [C | D] added to a CSS bivariate-bicycle stabilizer matrix: `H = (A B | C D; 0 0 | B^T A^T)`), mined from a dedicated weight-5-targeted non-CSS campaign rather than a fresh search here. This is this project's own headline non-CSS result (arXiv:2610.06623, "the best-connected PBB parameter point").
Source: github.com/qiskit-community/qcode-discovery @ d6abdbd, paper/2610.06623/weight5_paper.tex. A dedicated weight-5-targeted LLM-guided evolutionary search (OpenEvolve, "PBB-large" campaign, ell,m swept across six lattices) over 4-tuples (A, B, C, D), screened by k, isotropy, a BP-OSD distance estimate, and a combined non-CSS gate (evaluation/noncss_gate.py: verify_not_lc_css then is_equivalently_css on the constructed code — only candidates that are *not* CSS and *not* LC-equivalent to CSS under every tested Clifford class survive at all). Standout candidates were then deep-MILP-verified per logical (evaluation/distance_milp.py), and BLISS-deduped into equivalence classes.
The underlying code was first found as the connected half of a disconnected parent presentation at (ell=18, m=12), n=432, k=8 (two isomorphic components, evaluation/connectivity.py:stabilizer_components + components_isomorphic); the parent's exact MILP distance (all 16 logicals proved optimal at weight 10, results/weight5_pbb_large_v1_deep_milp_verify.jsonl) transfers to each component. A standalone, explicit presentation of one component at its native (ell=18, m=6) was then independently constructed and certified (results/weight5_pbb_component_certifications.jsonl, match W5PBB-216-4-10-36c5ce3b): re-run through the same non-CSS screens (commutation verified; css_equivalent_under_tested_classes: false under both the uniform 36-assignment brute force and the non-uniform {I,S}/ {H,HS} affine-system check) and confirmed connected.
Per-code model attribution (unlike the companion [[144,12,12]] submission from an earlier campaign where this wasn't tracked): the parent (18,12) presentation's code_key was recovered in this project's own per-program lineage log 15 times across the run, attributed to both aws/claude-opus-5 and azure/gpt-5.6-sol (attribution_status: "exact", recovery_method: "timestamp_or_generator_replay_join") — i.e. this is a genuine joint discovery, not a single-model find. A second, non-isomorphic construction at the same [[216,4,10]] point exists (parent `A = 1 + x^4y^2 vs. this one's A = 1 + x^2y^2, C = 0 vs. C = 1 + x^2y^2`; results/weight5_pbb_component_certifications.jsonl, match W5PBB-216-4-10-ae4cc6ad; same joint two-model attribution, 12 observations) and is not separately filed here since both occupy the same point in this board's Pareto space — noted here per the "one code, one entry" / alternate-construction convention, available on request if useful as a second filing.
(ℓ, m) = (18, 6), A = 1 + x^2y, B = 1 + xy^3 + x^2, perturbation `C = 1 + x^2y, D = 1 + x^2 (0-indexed exponent pairs: A=[[0,0],[2,1]]`, B=[[0,0],[1,3],[2,0]], C=[[0,0],[2,1]], D=[[0,0],[2,0]]). n = 2·18·6 = 216, k = 4, max check weight 5.
logicals optimal** at weight 10 on the (18,12) parent (logicals_optimal: 16/16, method "all-logical MILP optimum") — exact, transferred to this component by the isomorphism/homogeneity argument above.
check matrix directly from the (18,6) component's own generators via evaluation/pbb_code.py:build_pbb_code, then ran this repo's own cli/qldpc.py submit --dry-run (RIS search, 50,000 trials, gf2_fast accelerator at 2,000,000 trials). Independently found d<=10, matching the source's exact claim.
Final claim here: witness-backed upper_bound (per this board's confidence rules for stabilizer-type entries); the source repository's own evidence tier is exact.
Not applicable — this submission mined an existing, already-vetted and now-published campaign result rather than running a new search in this repository.
Model: jointly attributed to Claude Opus 5 and GPT-5.6 Sol (both aliases recovered against the parent code's lineage log, 15 independent observations across the run — see "What was searched" above), matches provenance.model. This submission's rebuild, independent re-verification, and packaging in *this* repository was done separately, using only this repository's own tooling (cli/qldpc.py submit, verify/).
from evaluation.pbb_code import build_pbb_code
code = build_pbb_code(ell=18, m=6,
A_terms=[(0,0),(2,1)],
B_terms=[(0,0),(1,3),(2,0)],
C_terms=[(0,0),(2,1)],
D_terms=[(0,0),(2,0)])
# code.matrix is the (216, 432) symplectic stabilizer matrix [X-part | Z-part]
Source: github.com/qiskit-community/qcode-discovery @ d6abdbd, evaluation/pbb_code.py, results/weight5_pbb_component_certifications.jsonl.