[[112,12,8]] weight-6 generalized bicycle code on Z_28 x Z_2

Direction & hypothesis

Track cell unrestricted x weight-6, small n (n <= 160). At k = 12 the board went [[72,12,6]], [[108,12,7]], [[126,12,10]], [[144,12,12]]: nothing with n < 126, k >= 12 and d >= 8. The designed-k mechanism (fix k by construction, spend the search on d) from the 2026-07-14 fieldnote was applied to abelian two-block group-algebra codes of weight 3+3: for abelian G the dimension is k = 2 dim(ker L(a) intersect ker L(b)), so kernels of all normalized weight-3 supports can be tabulated once per group and paired to hit k >= 8 exactly, with the surrogate spent only on pairs that could land on the frontier.

What was searched

a sweep script kept with the search run (sweep_designed.py, not committed; the method is described above), four runs on 2 threads: N = 72..80 (18360 pairs screened, 0 hits), N = 63..71 plus a partial Z_62 (about 9000 screened; Z_62 gave [[124,10,10]] 416 times and [[124,10,8]] 649 times, no d >= 11), N = 36..49 (23901 screened, 0 hits) and N = 50..61 (about 12000 screened before the budget stopped it). Groups: Z_N and Z_l x Z_m with gcd(l, m) > 1. Per group at most 400 supports with kernel dimension >= 4 and at most 3000 pairs with k >= 8, a nondomination threshold need >= 5, and a and b generating G (disconnected Tanner graphs are rejected by the verifier). Ladder 600 -> 5000 -> 20000 gf2_fast RIS trials per side, need enforced at each rung. On Z_28 x Z_2 (384 supports, 2664 pairs) 432 pairs reached [[112,12,8]]; on Z_14 x Z_4 339 pairs did; none reached d = 9.

Evidence trail

Ladder for the staged pair a = {(0,0), (1,0), (3,1)}, b = {(0,0), (2,0), (20,0)} (indices 0, 2, 7 and 0, 4, 40 in cyclic_product(28, 2) order): 600 -> 8, 5000 -> 8, 20000 -> 8, 200000 -> 8, 1000000 -> 8 trials per side, flat. The gate found no lighter logical in its own refutation pass and labelled the code "advances the weight-6 x unrestricted board". Witness-backed upper bound d <= 8 (X and Z witnesses both weight 8); not certified exact.

Fills the k = 12 gap between [[108,12,7]] and [[126,12,10]]; beats no frontier entry outright. It dominates the [[120,10,8]] found earlier in this run (Z_60, 2+4 split), which was therefore not staged.

Dead ends

• Random-support sweep (the random-support sweep, 401k samples): 91 percent have

k < 4; the connected survivors were the k = 10 fills [[124,10,10]] and [[120,10,8]]. A random [[112,12,8]] on Z_14 x Z_4 with A = {(0,0),(11,1),(6,2)}, B = {(0,0),(3,1),(2,2)} was disconnected (all of A and B lie in the index-2 subgroup x + y even): two copies of [[56,6,8]].

• Designed-k runs found nothing at N in 36..49 or 63..80: at n = 72..98 the

thresholds (d >= 7 at k = 12, d >= 8 at k = 10, d >= 9 at k = 8) were not reached by any of 23901 connected k >= 8 pairs, so [[72,12,6]] and [[90,10,7]] were not beaten by weight-6 abelian 2BGA in this sample.

• High kernel dimension for a weight-3 support usually means the support lies

in a proper subgroup: 60 to 85 percent of k >= 8 pairs were discarded as disconnected.

Tools

Claude Fable 5.1 (Claude Code, unattended workflow). research/kit (group_algebra, css, surrogate with gf2_fast at 2 threads, submit); gate verify/validate_candidate.py. About 25 CPU-minutes of screening across the designed-k runs, 40 s deep refutation for the finalist.

Reproduction

import sys; sys.path.insert(0, "research/kit") from group_algebra import cyclic_product, build_2bga mul, _ = cyclic_product(28, 2) HX, HZ = build_2bga(mul, [0, 2, 7], [0, 4, 40])

In polynomial form over F_2[x, y]/(x^28 - 1, y^2 - 1): a = 1 + x + x^3 y, b = 1 + x^2 + x^20, H_X = [A | B], H_Z = [B^T | A^T].