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"gf2_fast RIS in the packaging search; the ladder searched both sides jointly", "seeds": [ 7, 77001, 77138, 77275, 77412, 880001 ] } } }, "provenance": { "authors": [ "@dorakingx" ], "construction": "Open-boundary planar bivariate-bicycle code (Liang, Eberhardt, Chen, arXiv:2504.08887, Sec. II and Table V) with f = x + x^2 + y^2 and g = 1 + x^2 y + x^2 y^2 on a 16 x 16 grid (research/local2d/planar.py build_open_directional(16, 16), n = 2*16^2 = 512, k = 8), then reduced to n = 454 by 58 qubit removals of three kinds. (36) Restricted r=1 lattice grafting (Sec. III E): a qubit lying in exactly one stabilizer of some type is removed together with that stabilizer; accepted only if k stayed 8 and a bit-packed RIS search found nothing lighter than 17, screened at 20000 trials with a FIXED seed and then confirmed at a deeper rung, a qubit failing the confirm rung being blacklisted. Those in-loop rungs are a filter, not the evidence -- on other members of this family they let a unit of distance through -- so the claim below rests on the separate fresh-seed ladder. Its upper bound for this code equals the unreduced L = 16 code's own ladder bound, which is consistent with the reduction having lost nothing at this size but does not prove it: both are upper bounds, no lower bound exists for either, and a MILP asked whether any logical of weight below the claim exists hit its time limit with no verdict. (19) Weight-1 stabilizer cleanup, which runs no distance search at all because it needs none (Sec. III D step 4, research/local2d/boundary_engine.py _cleanup): a qubit carrying a weight-1 stabilizer cannot appear in any opposite-type check, so multiplying that row into the same-type rows containing it and deleting the qubit preserves k and the distance exactly; the elimination cascades until no weight-1 or zero-degree qubit is left. (3) Capped r=2 merge-graft, a new step: when a qubit lies in exactly two stabilizers R1, R2 of one type, replacing R2 by R1 + R2 is a row operation on the stabilizer generators and so changes no code, only the generating set, and it leaves that qubit in one stabilizer, where the r=1 graft above applies. The merge is taken only if R1 + R2 still has weight at most 6 and its support still has maximum pairwise distance at most 4 in the layout below -- exactly the quantity the verifier measures as the interaction radius -- so the weight class and the 2D-local single-layer class are preserved by construction; the resulting graft is then accepted under the same k check and the same screen and confirm rungs, and the whole layout radius is recomputed after every accepted move. The first two moves cannot enlarge a check support -- grafting deletes, and the cleanup XOR removes exactly the one fixed qubit from a row -- while the third can, and here did: one accepted merge replaced two weight-3 checks of diameter 2.2361 by a single weight-4 check of diameter 3.0000. The caps bound that growth rather than forbidding it, and the whole layout is re-measured after every accepted move, which is why the code below still has max check weight 6 and radius exactly 4. SINGLE-LAYER integer layout: a surviving qubit whose index in the unreduced code is q = c*256 + i*16 + j (block c in {0,1}, site (i,j)) sits at (i + j, j - i + c), the unit square lattice rotated 45 degrees with the two blocks on its two sublattices; one qubit per site, minimum spacing exactly 1, maximum check diameter exactly 4. The reduction is a randomised search, so the surviving qubit set is recorded by the coordinates in this file rather than re-derived.", "references": [ "arXiv:2504.08887", "arXiv:2308.07915" ], "date": "2026-09-09", "notes": "Checked, not equivalent to any board entry: the gate reports no exact duplicate and no WL-equivalent entry. The one board entry with n = 454 is [[454,2,33]] (no 2D layout), which differs in k and d. It dominates [[457,8,17]] on (n, k, d, w), which therefore leaves the frontier of every cell the two share. The ungrafted L = 16 member ([[512,8,...]]) is not on the board either. Both the construction and the grafting technique are published (arXiv:2504.08887, Sec. II, Sec. III D step 4 and Sec. III E; this repository's own research/local2d modules cite Table V of that paper for the family's ungrafted distances and Tables II-IV for its qubit-removal layouts such as [[188,8,9]], which come out of the cleanup rather than the graft), but this parameter set, the capped r=2 merge-graft step and this single-layer layout are not, so novelty is left unknown and literature novelty is unverified.", "model": "Claude Opus 5 (Claude Code)", "origin": "submission", "novelty": "unknown" }, "family": "bivariate-bicycle", "locality": { "coordinates": [ [ 0.0, 0.0 ], [ 1.0, 1.0 ], [ 2.0, 2.0 ], [ 3.0, 3.0 ], [ 4.0, 4.0 ], [ 5.0, 5.0 ], [ 6.0, 6.0 ], [ 7.0, 7.0 ], [ 8.0, 8.0 ], [ 9.0, 9.0 ], [ 10.0, 10.0 ], [ 11.0, 11.0 ], [ 12.0, 12.0 ], [ 13.0, 13.0 ], [ 2.0, 0.0 ], [ 3.0, 1.0 ], [ 4.0, 2.0 ], [ 5.0, 3.0 ], [ 6.0, 4.0 ], [ 7.0, 5.0 ], [ 8.0, 6.0 ], [ 9.0, 7.0 ], [ 10.0, 8.0 ], [ 11.0, 9.0 ], [ 12.0, 10.0 ], [ 13.0, 11.0 ], [ 14.0, 12.0 ], [ 16.0, 14.0 ], [ 2.0, -2.0 ], [ 3.0, -1.0 ], [ 4.0, 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