[[88,8,7]] — 2BGA on the dicyclic group Dic_11

Direction & hypothesis

Advance the unrestricted, weight-6 frontier with two-block group-algebra codes over non-abelian dicyclic (generalized-quaternion) groups Dic_m of order 4m, a family not represented on the board. Kept to moderate distance, where the RIS distance surrogate is reliable and the claim is verifiable.

What was searched

2BGA H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over small dicyclic and dihedral groups, random a, b with max check weight |a|+|b| <= 12, screened at 40k RIS trials, keeping only codes that strictly dominate a board entry on (n, k, d, w). This code: Dic_11 (order 44, so n = 88), a weight 4, b weight 2, giving max check weight 6 and k = 8.

Evidence trail

Witness-backed upper bound d <= 7, held under escalating RIS search rather than accepted at the screening budget: 40k trials -> 7, 300k -> 7, 1,000,000 (pair-depth 20) -> 7, no lighter logical found. The value is stable across the range, so 7 is a tight bound at this scale. The verifier accepts the code at kd^2/n = 4.455.

It strictly dominates these board codes (lower n, and/or higher k, and/or higher d, at no higher check weight): 128-8-6, 91-5-7, 112-6-7, 88-6-6.

Dead ends

High-distance variants of the same construction are not trustworthy on commodity hardware: at d ~ 30-40 the RIS surrogate over-estimates and collapses only under ~10^8-scale search. The search is confined to the low-to-moderate distance regime where the surrogate converges.

Model & harness

Found by Claude Opus 4.8 driving a dicyclic-2BGA cell-dominance search on the repo's gf2_fast RIS core; distances re-verified at 1M trials, then re-run through verify/qldpc_verify.py during packaging.

Reproduction

Group Dic_11 (order 44): a^{22} = 1, b^2 = a^{11}, b a b^-1 = a^-1; element (a^i b^j) at index 2i + j. Take a = [3, 5, 13, 43], b = [1, 21] as element indices and build the 2BGA via research/kit/products.lifted_product(mul, a, b) to obtain [[88,8,7]].