[[64,16,6]] — 2BGA on the metacyclic Z_16 x| Z_2 group

Direction & hypothesis

Advance the low-blocklength frontier with a two-block group-algebra code over a non-abelian group. Small n and modest distance were chosen deliberately: this is the regime where the RIS distance surrogate converges, so the claim is verifiable rather than an optimistic upper bound.

What was searched

2BGA H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over a roster of dicyclic, dihedral and metacyclic groups, sampling a, b with bounded check weight and keeping only codes that strictly dominate an existing board entry in their (locality, weight) cell. This code sits on the metacyclic Z_16 x| Z_2 group with max check weight 10.

Evidence trail

Witness-backed upper bound d <= 6, with an explicit weight-6 logical on each side. The code was rebuilt from its group data and re-witnessed independently of the search, and a deeper RIS pass returned the same distance rather than a lower one. verify/qldpc_verify.py accepts it at kd^2/n = 9.0. It strictly dominates 1 existing board entries.

Dead ends

The same search at high distance is not trustworthy: candidates screening near d ~ 40 collapsed under deeper search (one [[390,82,41]] screen resolved to 38 at 20M reads, matching the incumbent rather than beating it). Everything submitted here is confined to the low-distance regime where repeated deeper passes agree.

Model & harness

Found by a continual non-abelian 2BGA dominance search (Claude Opus 4.8) built on the repo's own research/kit group-algebra constructors and the gf2_fast RIS core; rebuilt, re-witnessed, and run through verify/qldpc_verify.py during packaging.

Reproduction

Build the group with research/kit/group_algebra (dicyclic presentation a^{2m}=1, b^2=a^m, bab^-1=a^-1 for Dic_m; metacyclic(p,k,r) otherwise), then build the 2BGA via build_2bga(mul, a, b) with the a, b element-index lists recorded in the code file to obtain [[64,16,6]].