[[60,4,8]] — weight-5 two-block group-algebra code on the non-abelian metacyclic group C_3 x| C_10, two-layer 2D-local layout

Direction & hypothesis

Target: the check-weight-5 slice of the weight-6 boards at small n. The board has no weight-5 cell; a weight-5 code competes on the weight-6 boards and holds a Pareto position through its lower weight. Before this campaign no code on the board with max check weight <= 5 exceeded kd^2/n = 4.00 (codes/40-10-4.json), and nothing in the 2D-local bilayer weight-6 cell with n <= 60 had k >= 4 and d >= 8. Hypothesis: the abelian weight-(3,2) family saturates near 4.75 (see the [[182,6,12]] note), and non-abelian groups might reach the same efficiency at smaller n.

What was searched

Two-block group-algebra (2BGA, Lin-Pryadko) codes H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with |a| = 3 and |b| = 2 (check weight 5) on 60 non-abelian groups of order <= 48: dihedral groups, all metacyclic groups C_n x| C_k with r^k = 1 mod n, S_4, A_4, and products of these with cyclic groups. Per group, 6,000 random (a, b) supports, deduplicated, k computed exactly, then 300 RIS trials per side (gf2_fast), ranked by kd^2/n. The winner, at 4.267, is this code on MC(3,10,2) = C_3 x| C_10 with the action i -> 2^j i mod 3; thirteen distinct (a, b) supports on that group give the same [[60,4,8]] parameters (presumably equivalent codes). The runner-up was [[96,4,10]] on C_3 x| C_16 at 4.17.

Evidence trail

Confirmation ladder (RIS trials per side -> lightest logical): 300 -> 8, 2,000 -> 8, 20,000 -> 8. The qldpc submit witness search (20,000 Python RIS trials plus a 2,000,000-trial accelerator pass) produced the witnesses in the JSON and found nothing lighter. At n = 60 this search depth is far beyond the fieldnotes' trial-depth floors. verify/validate_candidate.py passed: refutation held, not a duplicate or WL-equivalent of any board entry.

Claim, stated precisely: d <= 8 on both sides is a **witness-backed upper bound**. An exact certificate (verify/certify.py) is cheap at this size and is the natural next step; it has not been run for this entry.

Dead ends

• Weight-4 2BGA codes: a 77-group non-abelian pilot (dihedral to D_32, all

metacyclic groups of order <= 64, S_4, A_4, D_k x Z_m; 4,000 supports each) never exceeded kd^2/n = 2.000, matching the abelian bound (a weight-4 abelian two-block code is a disjoint union of toric codes, so kd^2/n <= 2).

• Weight-5 non-abelian groups sit on the same plateau as the abelian ones

(4.0 to 4.8): the non-abelian structure buys smaller n, not higher efficiency.

• A single-layer layout (r <= 4.0) was not found; the annealer floors near

r = 5 with one layer for the codes of this size.

Tools

Claude Fable 5.1 driving Claude Code, one lane of a five-agent campaign with a shared append-only notes file. Repo tooling: research/kit/group_algebra.py (metacyclic, build_2bga), research/kit/css.py, the gf2_fast RIS backend via research/kit/search.py, research/local2d/fold_layout.py::anneal (2 layers, integer grid sites, boxes 6x5, 8x4, 7x5, 6x6 with 3 seeds each; best r = sqrt(13) = 3.606 in the 6x6 box), research/kit/submit.py, verify/validate_candidate.py. About 20 core-minutes for the pilot, under a minute for the layout.

Reproduction

mul = research.kit.group_algebra.metacyclic(3, 10, 2) (element (i, j) at index i*10 + j, identity at 0). `HX, HZ = build_2bga(mul, a=[0, 18, 20], b=[0, 3])`, i.e. a = {g_0, g_18, g_20} and b = {g_0, g_3} as element indices, H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T]. Then n = 60, k = 4, every row has weight 5. Layout: fold_layout.anneal(checks, 60, box_sites(6, 6), layers=2); measured interaction radius sqrt(13) = 3.606, at most 2 qubits per site, unit spacing.