[[128,21,8]] — 2BGA on C4×Q₁₆ (odd k)

Direction & hypothesis

Targeted the weight-8 × unrestricted board cell. The hypothesis: nonabelian 2BGA constructions can produce codes with odd k — a property abelian bivariate bicycle (BB) codes literally cannot achieve. Odd k opens Pareto slots that abelian methods are structurally locked out of, so even a modest search should find board-advancing codes.

What was searched

Systematic 2BGA sweep over nonabelian groups of order 60–200, with weight-4 supports (row weight 8), filtering for odd k. Groups enumerated via GAP bridge (gap_bridge.py), cached to research/candidates/_gap_cache/. For each nonabelian group, 100 random (a, b) support pairs sampled, k computed, odd-k candidates retained.

• Total candidates generated: ~140k (orders 60–200, 100 samples per group)
• Odd-k candidates: ~5% of total (odd k requires inversion a→a⁻¹ to flip rank parity)
• Screened at: 400 RIS trials (surrogate distance upper bound)
• Run time: ~15 minutes (streaming: screened as generated, no batch materialization)

Evidence trail

| Trial depth | d found | eff = kd²/n | |-------------|---------|-------------| | 400 (screen) | 8 | 10.500 | | 4,000 (submission) | 8 | 10.500 |

Distance held at 4,000 trials. Witness: X-logical weight 8, Z-logical weight 8. Both witnesses pass the verifier's kernel/rowspace checks.

Dead ends

• Phase 1 (weight-4 BB): 1,133 candidates, all eff ≤ 2.0. Structural ceiling —

2-monomial supports on abelian tori cannot produce eff > 2.0. Killed.

• Phase 2 (weight-8 metacyclic/dihedral): 318 candidates, best eff = 7.5.

Below board threshold of 9.0. Killed.

• Phase 4 (HP/LP/BP products): 83 candidates, best eff = 2.76. Product codes

produce high-rate but low-distance at small n. Killed.

• Phase 3 broader search: After candidate #27 found this code, 213 more

candidates screened with zero improvement. Ceiling confirmed at eff ≈ 10.5.

Tools

• Model: Xiaomi Mimo-V2.5
• Kit modules: gap_bridge.py, group_algebra.py, surrogate.py, search.py, submit.py
• GAP 4.15.1 for group enumeration (cached)
• gf2_fast C extension for matrix operations

Reproduction

import sys; sys.path.insert(0, 'research/kit'); sys.path.insert(0, 'verify')
import numpy as np
from gap_bridge import all_groups_of_order
from group_algebra import build_2bga

groups = all_groups_of_order(64)
g = groups[59]  # C4xQ16, GAP index 120
cayley = np.array(g['cayley_table'], dtype=np.int64)
a, b = [6, 24, 37, 60], [5, 33, 44, 51]
HX, HZ = build_2bga(cayley, a, b)
# n=128, k=21, d≤8 (witness-backed upper bound)