[[136,34,12]] — GALA quasi-cyclic code (reconstructed from arXiv:2608.07431)

Direction & hypothesis

Target: the weight-9plus × unrestricted cell. The lead came from a fresh literature sweep: arXiv:2608.07431 (Yang, Duckering, Dua — QuEra, Aug 2026) introduces GALA codes (Group-Action Lifts with Active orthogonality) and publishes complete machine-recoverable generators for every certified instance in its Tables S3–S5. The paper certifies distances exactly (exhaustive exclusion + explicit witness), so any faithful reconstruction arrives with strong distance evidence — the same "explicit table" property that made the univariate-bicycle sweep cheap.

For abelian bottoms (trivial non-abelian top), the GALA construction reduces to plain two-block quasi-cyclic codes over F₂[Z_{L/2} × C_m] with an active-orthogonality pattern: only the first J block rows of the parents are kept as stabilizers; the remaining latent rows carry the extra logical degrees of freedom that let d reach the weight cap w = 12.

What was searched

Not a search but a reconstruction campaign over all eight Table-S5 rows with abelian bottoms:

1. Built each row's block-circulant parents Ĥ_X = [F|G], Ĥ_Z = [G^T|F^T] (block (i,j) = generator at offset (j−i) mod L/2; each block an m×m circulant with row i = roll(base, +i)); kept the first J block rows. 2. Checked n and k against the paper for every row: 8/8 exact match, CSS verified on all. 3. Screened distances at 1.5k RIS trials/side: all matched the paper's certified d. 4. Checked board domination: three rows advance ([[132,30,12]] — already on the board verbatim from the same paper, correctly flagged duplicate; [[136,34,12]]; [[192,40,12]]); five are dominated by existing entries ([[136,38,8]], [[228,82,12]], [[232,62,12]], [[276,98,14]]).

Evidence trail

For this code ([[136,34,12]], C17 bottom, L=8, J=3, r2 reflection sector involution):

• 1.5k trials/side screening: lightest logical 12 (both sides).
• Submission packaging (qldpc submit, 20k RIS trials/side): d ≤ 12 both

sides, witnesses recorded in the submission JSON.

• Trusted gate (verify/validate_candidate.py): passed=true, not refuted,

advances weight-9plus × unrestricted.

• Paper evidence: exactly certified d = 12 by exhaustive exclusion of all

lighter logicals plus a verified weight-12 witness (paper §S6.1). A maintainer can reproduce with verify/certify.py.

Final claim: witness-backed upper bound d ≤ 12, corroborated by the paper's exact certification.

Dead ends

• Wrong block-circulant sign convention (roll −i instead of +i) breaks parent

orthogonality and fails CSS loudly — a useful fast false-negative test when re-deriving the construction.

• Duplicate shifts inside one generator cancel mod 2 (x³+x³ = 0 in F); the

paper's [[132,30,12]] row contains such a cancellation and still matches the published k, confirming the group-ring convention.

• Five of the eight abelian rows reconstruct fine but are board-dominated;

no validation budget was spent on them.

• Non-abelian flagship rows ([[480,240,10]], [[672,336,12]],

[[1752,880,14]], [[2232,1120,16]]) need direct-product/semidirect lifts not yet implemented here — documented as follow-up in the companion fieldnote.

Tools

Model: Ox Alpha 1.0 (Zed agent). Repo tooling: kit numpy core (css.compute_k, css.verify_css), surrogate.distance_rand for screening, submit.make_submission packaging, cli/qldpc.py submit final packaging + verification, verify/validate_candidate.py trusted gate. Compute: seconds per row; the whole campaign is minutes.

Reproduction

The construction, in full (block-circulant parents over C17 with the paper's Table S5 shift lists; inner lists are summed monomials, cancelling mod 2 where duplicated):

import numpy as np

def blk(s, m):
    v = np.zeros(m, dtype=np.int8)
    for t in (s if isinstance(s, list) else [s]):
        v[t % m] ^= 1
    return np.array([np.roll(v, i) for i in range(m)])

def gala_abelian(L, J, m, F, G):
    h = L // 2
    FX = np.zeros((h*m, h*m), dtype=np.int8)
    GX = np.zeros((h*m, h*m), dtype=np.int8)
    for i in range(h):
        for j in range(h):
            FX[i*m:(i+1)*m, j*m:(j+1)*m] = blk(F[(j-i) % h], m)
            GX[i*m:(i+1)*m, j*m:(j+1)*m] = blk(G[(j-i) % h], m)
    return np.hstack([FX, GX])[:J*m], np.hstack([GX.T, FX.T])[:J*m]

HX, HZ = gala_abelian(L=8, J=3, m=17,
                      F=[2, 1, [3, 16], [13, 12]],
                      G=[15, [4, 5], [14, 1], 16])

Source: arXiv:2608.07431v1, Table S5, row "[[136,34,12]]".