n
240
Cell weight-6 x unrestricted, CSS board, n between 221 and 300. The weight-6 frontier there ran 224-18-12, 224-12-16, 234-26-7, 240-16-14, 254-14-16, 270-30-7, 288-16-16: at n <= 240 the largest k with d >= 8 was 18 (224-18-12), and k >= 24 only existed at d <= 7 (234-26-7). The hypothesis was that two-block group-algebra (2BGA) codes on non-abelian groups of order 111 to 150 could put a k = 24, d = 8 point into that corner at check weight 6. [[240,24,8]] does: no board entry has n <= 240, k >= 24, d >= 8 and check weight <= 6, and the k - 2 margin holds as well ([[240,22,8]] is not dominated either).
Random-support 2BGA codes H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with a, b three group elements each, both containing the identity (no loss of generality: multiplying a or b by a group element permutes qubits and checks). The group menu was every non-abelian group of order 111 to 150 the kit builds from Cayley-table constructors: dihedral D_n, metacyclic C_n ⋊_r C_k (up to two actions r per (n, k)), C_m x D_k, C_m x A4, C_m x S4, C_m x A5, D_a x D_b, C_m x (C_n ⋊ C_k) for eight small metacyclic factors, S5, and the split metacyclic presentations ZSZ(l1, l2, q) of research/kit/nonabelian_lp.py.
Random supports almost never give usable k (a generic weight-3 element is invertible on the higher-dimensional irreducible blocks of F_2[G]), so the sweep ran a parallel generate-and-rank prefilter: 400,000 draws at 222 <= n <= 300 (seed 61), of which 519 had k >= 16 and were connected. Those 519 went through research/kit/search.screen with the gf2_fast backend at 400 trials; 8 advanced the board frontier at screen depth and were confirmed at 20,000 / 200,000 / 2,000,000 fast trials (research/kit/surrogate.distance_rand_witness, pair depth 8), dropping a candidate as soon as its running bound stopped advancing. Besides this code the survivors were a [[270,24,8]] on ZSZ(45,3,16) (dominated by this one) and six draws of a [[252,20,10]] on presentations of C_21 x S_3 (filed separately).
Witness-backed upper bound d <= 8, not an exact distance. The bound was 8 at 400 trials and never moved; both sides have a weight-8 logical.
| stage | trials per candidate | lightest logical | |---|---|---| | screen (gf2_fast, seed from fingerprint) | 400 | 8 | | confirm | 20,000 | 8 | | confirm | 200,000 | 8 | | confirm | 2,000,000 | 8 | | numpy lightest_logical, both sides, seeds 61 / 62 | 1,500 | X 8, Z 8 | | deep ladder, 3 fresh seeds (48247, 56166, 64085), gf2_fast | 10,000,000 each | 8, 8, 8 (X side) |
verify/validate_candidate.py (through research/kit/coordination.py's gate wrapper) returned passed: true, board_advancing: true, `d_only_gain: false`, label "advances the weight-6 x unrestricted board". Literature novelty is unverified; board-relative novelty comes from verify/validate_candidate.py (fingerprint and Weisfeiler-Leman dedup against every board entry found nothing).
check weight 6 (weight-3 a and b): at odd m from 111 to 200 and deg g >= 8 there are essentially no weight-3 multiples of g; the few that exist are lifts from a subring Z_{m/s} and give direct sums.
prefiltered draws (2,881 with k >= 18) across 222 <= n <= 400: every screen-depth survivor collapsed by 20,000 trials (dominated by 248-24-18, 252-20-21, 252-26-18, 292-22-23, 336-24-24, 378-18-27 and similar).
serial draws found nothing; 200,000 prefiltered weight-6 draws (185 with k >= 12) gave only points dominated by the 2BGA finds above.
(check weight 8), 5,000 draws, nothing advancing; weight-(2,2) entries (check weight 6) give k = |G| + 6 at d = 6, already dominated by the board's 258-88-6.
S4 x C_m: 3,000 serial draws, nothing advancing.
Model: Claude Fable 5.1 (Claude Code agent, contributor-driven workflow). Compute: one Brev n2d-highcpu-64 box; gf2_fast at 30 threads for the confirmation rungs and the ladder, 30 forked processes for the generate + rank prefilter. Kit modules: research/kit/nonabelian_lp.py (zsz), research/kit/group_algebra.py (metacyclic, build_2bga), research/kit/search.py (screen), research/kit/surrogate.py (distance_rand_witness, lightest_logical), research/kit/prescreen.py (direct-sum and board dedup), research/kit/submit.py, research/kit/coordination.py. About 0.3 CPU-hours on this code; about 10 CPU-hours on the sweeps above.
The group is research/kit/nonabelian_lp.py's zsz(15, 8, 11), i.e. research/kit/group_algebra.py's metacyclic(15, 8, 11): C_15 ⋊ C_8 with y x y^-1 = x^11, element x^i y^j at index 8 i + j, identity at 0 (11 has order 2 mod 15, so y^2 is central and the group has order 120). With x = (1, 0), y = (0, 1):
a = 1 + x^5 y + x^10 y^6 (indices 0, 41, 86), b = 1 + x^7 + x^9 (indices 0, 56, 72).
import sys; sys.path[:0] = ["research/kit", "verify"] from nonabelian_lp import zsz from group_algebra import build_2bga from css import compute_k mul = zsz(15, 8, 11) HX, HZ = build_2bga(mul, [0, 41, 86], [0, 56, 72]) # n = 240, k = 24 assert compute_k(HX, HZ) == 24
Check sets match codes/240-24-8.json row for row in this order. Witness search: `surrogate.distance_rand_witness(HX, HZ, trials=2_000_000, seed=61000003 + 2_000_000, backend="fast", pair_depth=8)` returns a weight-8 logical; the numpy lightest_logical at 1,500 trials with seeds 61 (X) and 62 (Z) gives the two witnesses in the JSON.