n
108
Track cell: unrestricted x weight-any (max check weight 10). The board is dense below n = 110, so the target was a specific hole: a point with n' <= 108, k' >= 32, d' >= 6 at check weight <= 10 that no board entry occupies. This code gains on the cell frontier against the entries nearest to it. The hypothesis behind the sweep was that the small-group two-block (2BGA) family, enumerated over the groups of order 8..55 the kit can build (abelian, metacyclic, dicyclic, generalized dihedral, a few semidirect and matrix groups), still holds non-dominated points that the board's generalized-bicycle and bivariate-bicycle entries missed, because the board's small-n entries come mostly from cyclic groups and tori, and because unequal support sizes (|a| != |b|) are rarely searched. Board-relative novelty is from verify/validate_candidate.py; literature novelty is unverified (Lin-Pryadko's 2BGA tables, arXiv:2306.16400, cover many small groups and this instance may be in them).
H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over F_2[G] with G = the direct product of [the generalized dihedral group Dih(A) = A x| C_2 with C_2 acting by inversion, A = Z_{3}} x Z_{{3}; elements (t, j) with t in itertools.product order over A and j in {0,1}, index = 2*pos(t) + j; (t,j)(u,l) = (t + (-1)^j u, j + l)] with Z_{3} via group_algebra.direct_product(base, cyclic_product(3)) (index i*|C| + j). a = [0, 6, 7, 26, 48] and b = [0, 21, 38, 49, 51] are lists of element indices (|a| = 5, |b| = 5, check weight 10).
Sweep (research/kit samplers and surrogate.distance_rand, backend gf2_fast, one thread per candidate, 60 processes):
Supports were drawn with the identity in both a and b (left/right translation of a or b is a qubit relabelling, so this loses nothing), distinct elements only, and every candidate was compared with the live board (every CSS entry with n <= 130, on n, k, d and max check weight) before any further trials were spent. The screening reading is an upper bound, so this funnel never discards a true advance.
Packaging witnesses: surrogate.lightest_logical, 8,000 NumPy trials per side, seed 11 -> X-logical of weight 6, Z-logical of weight 6; d = min = 6.
Deep ladder (fresh seeds, surrogate.distance_rand_witness backend gf2_fast, 8 threads, pair depth 8):
| trials per seed | seed | side of lightest | lightest logical | time | |---|---|---|---|---| | 10,000,000 | fast seed 60010 | X | 6 | 50.4 s | | 10,000,000 | fast seed 67929 | X | 6 | 50.6 s | | 10,000,000 | fast seed 75848 | X | 6 | 50.6 s |
Ladder verdict: corroborated (min over rungs = 6, claimed 6).
Final claim: witness-backed upper bound d <= 6 (confidence: upper_bound), with d_X <= 6 and d_Z <= 6.
above): a d read at 400 fast trials is routinely 1-3 too high at n ~ 60-110.
weight-6 cell: the board's BB coverage there is effectively complete.
Model: Claude Opus 5.5 (agent harness Claude Code), running the search on a Brev n2d-highcpu-64 box (gf2_fast via make fast, 8 threads for the ladder, single-thread worker processes for the screen). Kit modules: research/kit/group_algebra.py (build_2bga, cyclic_product, metacyclic, direct_product, sym, alt), research/kit/gf2poly.py, research/kit/products.py, research/kit/surrogate.py, research/kit/prescreen.py, research/kit/submit.py, research/kit/coordination.py (gate). Approximate compute: about 3 CPU-hours for the sweep and ladder of this family.
import sys; sys.path[:0] = ["research/kit", "verify"] from group_algebra import build_2bga mul = <Cayley table of prod(gendih(3,3),(3,)) as described above, identity at index 0> HX, HZ = build_2bga(mul, [0, 6, 7, 26, 48], [0, 21, 38, 49, 51]) # n = 108, k = 32
Then make_submission(HX, HZ, ..., trials=8000, seed=11) from research/kit/submit.py reproduces the witnesses; the deep ladder used three fresh seeds at the trial counts in the table above.