[[78,6,11]] — 2BGA code on ab(13,3) (|G| = 39), a = [0, 4, 10, 12], b = [0, 3, 14, 38]

Direction & hypothesis

Track cell: unrestricted x weight-8 (max check weight 8). The board is dense below n = 110, so the target was a specific hole: a point with n' <= 78, k' >= 6, d' >= 11 at check weight <= 8 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).

What was searched

H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over F_2[G] with G = the abelian group Z_{13}} x Z_{{3} built as group_algebra.cyclic_product(13,3): elements are the tuples in itertools.product order, index = position in that order (identity = 0). a = [0, 4, 10, 12] and b = [0, 3, 14, 38] are lists of element indices (|a| = 4, |b| = 4, check weight 8).

Sweep (research/kit samplers and surrogate.distance_rand, backend gf2_fast, one thread per candidate, 60 processes):

• 2bga: 9,400,000 candidates built, 2,932,218 with k >= 3 screened at 400 gf2_fast RIS trials, 3893 unique records non-dominated on the board at that reading; funnel 20000 trials: 3893 of 3893 still advancing, 200000 trials: 3893 of 3893 still advancing, 200000 trials: 42 of 42 still advancing, 2000000 trials: 42 of 42 still advancing.

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.

Evidence trail

Packaging witnesses: surrogate.lightest_logical, 8,000 NumPy trials per side, seed 11 -> X-logical of weight 11, Z-logical of weight 11; d = min = 11.

Deep ladder (fresh seeds, surrogate.distance_rand_witness backend gf2_fast, 10 threads, pair depth 8):

| trials per seed | seed | side of lightest | lightest logical | time | |---|---|---|---|---| | 10,000,000 | fast seed 54957 | X | 11 | 27.7 s | | 10,000,000 | fast seed 62876 | X | 11 | 27.0 s | | 10,000,000 | fast seed 70795 | X | 11 | 28.4 s |

Ladder verdict: corroborated (min over rungs = 11, claimed 11).

Final claim: witness-backed upper bound d <= 11 (confidence: upper_bound), with d_X <= 11 and d_Z <= 11.

Dead ends

• Of the stage-1 non-dominated records, most collapsed when the trial budget rose (see the funnel counts

above): a d read at 400 fast trials is routinely 1-3 too high at n ~ 60-110.

• Abelian 2BGA (bivariate bicycle) supports at 3+3 on l*m in 32..55 reproduced board points but opened no new

weight-6 cell: the board's BB coverage there is effectively complete.

• Weight-4 (2+2) supports on non-abelian groups never beat the toric-code line kd^2/n = 2.

Tools

Model: Claude Fable 5.1 (agent harness Claude Code), running the search on a Brev n2d-highcpu-64 box (gf2_fast via make fast, 10 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.

Reproduction

import sys; sys.path[:0] = ["research/kit", "verify"]
from group_algebra import build_2bga
from group_algebra import cyclic_product
mul, _ = cyclic_product(13,3)
HX, HZ = build_2bga(mul, [0, 4, 10, 12], [0, 3, 14, 38])   # n = 78, k = 6

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.