[[390,78,9]] non-abelian lifted product over ZSZ(39,2,14) at check weight 8

Direction & hypothesis

Target: the unrestricted x weight-8 cell at rate 1/5. Before this submission the cell had no code with k >= 70 and d >= 9 below n = 392; the nearest points are [[232,62,12]] and [[392,102,14]]. 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, which brings the check weight to 8. The bet was that the high-k region of the weight-8 cell was empty enough that a d = 9 rate-1/5 code lands on its frontier.

What was searched

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|.

• 12000 random codes on 117 ZSZ groups with 12 <= |G| <= 60 at 500 fast

RIS trials: 6405 distinct with k >= 4, d >= 4; d = 9 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]].

• 6000 random codes on 291 ZSZ presentations with 61 <= |G| <= 140 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. From n = 480 up the points are dominated by [[472,122,16]] and [[488,126,16]].

• 6000 random codes on 138 small non-abelian groups at 300 trials: 3079

distinct, best 300:9, 420:10, 480:10, 600:11, 660:10, none better than the ZSZ points.

• Seed-pipeline variant (800 classical seed rows per group ranked by RIS

distance, top 12 x 12 products), 7 groups with |G| in 20..60 before it was stopped: best 100:5, 120:7.

Screening used research/kit/search.py 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 with 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.

Evidence trail

Submitted code (ZSZ(39,2,14), from the 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 559191786) | 8000 numpy RIS | nothing lighter |

Both witnesses in the JSON have weight 9. The claim is a witness-backed upper bound d <= 9; no exact certification was attempted (k = 78 is far outside the MILP envelope in CONTRIBUTING.md).

Frontier caveat, stated plainly: [[392,102,14]] (check weight 8) has more logical qubits and a larger distance at two more physical qubits. [[390,78,9]] is on the weight-8 Pareto frontier by n alone against that entry; against everything at n <= 390 it is the first weight-8 code with k >= 70 and d >= 9.

The same sweep's other ladder candidates that still advanced after the ladder all read flat from screen through 100k trials: [[350,70,9]] (submitted separately), [[360,74,8]], [[360,72,8]], [[320,64,8]], [[330,66,8]], [[330,68,7]], [[390,80,7]] and seven rate-1/5 points at d = 5 that are non-dominated only for lack of high-k weight-8 board codes; those were not packaged. On this group, ZSZ(39,2,14), the sweep also produced [[390,80,7]] and [[390,84,5]], whose extra logical qubits come from rank-deficient base rows and whose distance is lower.

Dead ends

• Profiles with an all-weight-2 side ((3,3)/(2,2), (3,2)/(2,2),

(4,2)/(2,2), (2,2)/(2,2)): the quantum RIS bound never exceeded the classical distance of the binomial seed row (150 of 150 random codes); that distance is the Cayley-graph girth of the row's two generators, <= 6 for every non-abelian ZSZ group with |G| < 105 and <= 8 up to |G| = 140. Capped at d <= 6 for n < 525.

• ZSZ(15,2,11), the group of the published [[150,30,10]], has no usable

weight-2 product: all 24 girth-6 generator pairs are related by a conjugate-shifted-inverse map that forces a weight-3 logical.

• Rate 2/5 with entry weights (2,2,2)/(2,2,2) at check weight 8: 6000

codes on 144 groups, all d <= 5.

• 2x3 monomial A with a (3,3) row B at check weight 8: 6000 codes on 201

groups, d 10 to 12 only at n >= 320, all dominated.

Tools

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.

Reproduction

Group G = ZSZ(39, 2, 14): generators x, y with x^39 = y^2 = 1 and y x = x^14 y; element x^a y^b at index 2a + b (|G| = 78, identity at 0).

Base rows (entries in F_2[G]):

A = [ 1 + x^11 y + x^36 , 1 + x^25 ] B = [ 1 + x^27 y + x^34 , 1 + x^14 ]

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 78 qubits, n = 5 x 78 = 390. 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, 23, 72], [0, 50]] and B = [[0, 55, 68], [0, 28]]. The same convention rebuilds codes/150-30-10.json from the trinomials in notes/150-30-10.md.