n
350
Target: the unrestricted x weight-8 cell at rate 1/5, between [[232,62,12]] and [[392,102,14]], where the board had no code with k >= 70 and d >= 9 below n = 392. The construction is the lifted product of two one-row base matrices with entries in F_2[G] for a non-abelian group G, the weight-9 mitten / ZSZ-LP shape of arXiv:2607.28795 and arXiv:2607.27644, with one entry per row lowered from weight 3 to weight 2. That drops the check weight from 9 to 8 at the cost of distance; the hypothesis was that at rate 1/5 the weight-8 cell forgives the loss, since its high-k region was empty.
Sampler: random supports of a prescribed entry-weight profile over a random group from the pool; entries of weight >= 2 contain the identity (no loss of generality for one-row bases). Groups: every non-abelian metacyclic presentation ZSZ(l1, l2, q) = Z_l1 x|_q Z_l2 (relation y x = x^q y, q^l2 = 1 mod l1, q != 1) with l2 <= 8, plus A4, S4, A5, C_m x A4, C_m x S4 and C_m x D_k.
Profile (3, 2) / (3, 2): A = [a_1, a_2] with weights 3 and 2, B likewise, check weight 8, n = 5|G|, k >= |G|.
500 fast RIS trials: 6405 distinct codes with k >= 4 and d >= 4; d = 9 reached only at n = 300 (3 of 1140 codes there), d = 8 at n = 160 to 300. Every point with k <= 62 and d <= 12 is dominated by [[232,62,12]].
screened at 300 trials: 4095 distinct, 602 passing the board pre-check at screen depth, best screen d by n: 350:9, 390:9, 480:10, 525:11, 600:11, 625:12, 700:11. Points from n = 480 up are dominated by [[472,122,16]] and [[488,126,16]].
direct products), 300 trials: 3079 distinct, best 300:9, 420:10, 480:10, 600:11, 660:10, all dominated or not better than the ZSZ points.
the top 12 x 12): stopped after 7 groups with |G| in 20..60, best 100:5 and 120:7, nothing beyond the random sweep.
Screening used the kit's research/kit/search.py screen with the fast RIS backend (verify/gf2_fast), dedup by rref fingerprint, then the board's (n, k, d, w) Pareto rule against codes/*.json (equality on all four axes counted as dominated). Ladder: 10k then 100k fast trials on the best d per (n, k) among pre-check survivors, at most 15 per sweep.
Submitted code (ZSZ(35,2,29), 61 <= |G| <= 140 sweep):
| stage | trials per side | lightest logical | |---|---|---| | screen | 300 | 9 | | ladder | 10k | 9 | | ladder | 100k | 9 | | packaging, round 1 (seed 7919) | 1,000,000 | 9 (X) | | packaging, round 2 (seed 15838) | 1,000,000 | 9 (X) | | gate refutation (seed 1065517832) | 8000 numpy RIS | nothing lighter |
Both witnesses in the JSON have weight 9 (X: 9, Z: 9). The claim is a witness-backed upper bound d <= 9; no exact certification was attempted (k = 70 puts a MILP certificate far outside the envelope described in CONTRIBUTING.md).
The other 14 ladder candidates from the same sweep that still advanced a cell after the ladder all read flat from screen through 100k trials ([[390,78,9]], submitted separately; [[360,74,8]], [[360,72,8]], [[320,64,8]], [[330,66,8]], [[330,68,7]], [[390,80,7]] and seven d = 5 points). The d = 5 points are non-dominated only because no w <= 8 board code has k that large at d = 5 and were not packaged.
(4,2)/(2,2), (2,2)/(2,2)): the quantum distance never exceeded the classical distance of the binomial seed row (150 of 150 random codes), and that distance is the Cayley-graph girth of the row, which is <= 6 for every non-abelian ZSZ group with |G| < 105 and <= 8 up to |G| = 140. Capped at d <= 6 for n < 525.
girth-6 generator pairs are conjugate-shifted-inverse related, which forces a weight-3 logical.
codes on 144 groups, all d <= 5.
6000 codes on 201 groups, best d 10 to 12 at n >= 320, all dominated.
by the two papers' codes.
Model: Claude Fable 5.1 (Claude Code), matching provenance.model. Repo tooling: research/kit/search.py (screen, fast backend), verify/gf2_fast for the 1M-trial two-sided RIS searches, research/kit/submit.py (make_submission, save_submission) for packaging with the numpy witness search, verify/validate_candidate.py as the gate. The lifted-product constructor and sampler were written for this run and are submitted to the research kit in a separate PR. The whole campaign (17 sweeps, about 67000 screened codes) ran in about five hours of wall clock on a 16-core machine; each 1M-trial round takes a few minutes per side with 16 threads.
Group G = ZSZ(35, 2, 29): generators x, y with x^35 = y^2 = 1 and y x = x^29 y; element x^a y^b at index 2a + b (|G| = 70, identity at 0).
Base rows (entries in F_2[G]):
A = [ 1 + x^14 + x^15 y , 1 + x^2 ] B = [ 1 + x^12 + x^28 y , 1 + x^32 ]
Entries of A act by the left regular representation L(g)[gh, h] = 1, entries of B by the right regular representation R(g)[h, hg] = 1. Sector 1 holds blocks (i, j), i in cols(A), j in cols(B), at block index 2i + j; sector 2 holds the single block (0, 0) after them; every block has 70 qubits, n = 5 x 70 = 350. X-check block row j: L(A[0][i]) on block (i, j) for i = 0, 1 and R(B[0][j]) on the sector-2 block. Z-check block row i: R(B[0][j])^T on block (i, j) for j = 0, 1 and L(A[0][i])^T on the sector-2 block. In group-element indices, A = [[0, 28, 31], [0, 4]] and B = [[0, 24, 57], [0, 64]]. The same convention rebuilds codes/150-30-10.json from the trinomials in notes/150-30-10.md.