[[144,76,4]] - Cornucopia code at q=4, a new member of the published family

Direction & hypothesis

Target: the weight-9plus x unrestricted cell on the CSS board (checks of weight 12, no 2D-local layout). The Cornucopia architecture paper (arXiv:2610.19073) closes by saying its construction "carries over to other codes of the family", and the family (Lu, Li & Deng, arXiv:2608.02773) is n = 36q, k = 18q + 4 with weight-12 checks - but the paper publishes instances only at q = 7, 16, 25, 29, 49, 64, 79, of which only q=7 and q=16 fit this board's n <= 700 cap and both are already board entries. So the unpublished block sizes under the cap, q = 4..19, were the opening: every one of them is a rate-1/2 point no CSS board entry can dominate (an entry dominating [[144,76,d]] would need n <= 144 and k >= 76, i.e. rate at least 0.528 at that size, and the board has none).

What was searched

The family's only free parameters are its 12 column shifts: check matrices HX = [A_(j-i) | B_(j-i)], HZ = [B^-1_(i-j) | A^-1_(i-j)] over Z_3 x Z_q coordinate permutations (row maps r+1 for A0 and B2, 2r+2 for A1 and B3, identity otherwise). CSS commutation holds for *every* shift choice: the only noncommuting row-map pairs, (A0,B3) and (A1,B2), enter block-row differences i'-i = 3 (mod 6), which never occur for i,i' in {0,1,2}; so the search is over shifts alone, with k recomputed exactly on every draw.

At q=4: 1200 uniformly random shift pairs, screened at 100 RIS trials each (accelerated backend); the top 24 re-measured at 4000; the top 4 re-measured at 100000; then the packaged candidate re-verified by submit.make_submission (8000 numpy trials/side) and by the submission CLI (800 python RIS + 300000 accelerated trials/side). Stage-1 reads at q=4: 2 -> 909, 4 -> 289. Seed 17 throughout the sweep, 0 for the CLI witness search. One design fact from the sweep: a read is the lightest logical *found*, so it is always >= the true d - stage-1 ranking cannot discard a better code, only fail to see how good it is.

Evidence trail

Ladder for this candidate (fast-RIS read -> numpy witness -> CLI):

• stage reads at 100, 4000, 100000 fast-RIS trials: 4, 4, 4; packaging numpy witness search: 8000 trials/side -> d <= 4.
• final claim: witness-backed upper bound d <= 4, not exact (the

exact tier needs the server MILP; k = 76 puts this beyond its envelope).

• k = 76 = 18*4+4 recomputed by the verifier; CSS commutation checked;

max check weight 12.

• Gate (verify/validate_candidate.py): passed: true, refuted: false

(its own refutation found no lighter logical), board_advancing: true, d_only_gain: false, dominated_by: [].

• The 1200 draws at this q never read above 4; deeper stages confirmed

4. kd^2/n = 8.44.

Dead ends

• ~0.2% of random shift sets drop rank (k < 18q+4): the two

redundancies per check type are not always the only ones. Rejected on an exact GF(2) rank, not a heuristic. At q=4: 2 of 1200 draws.

• At q=7 and q=16 the best random-shift reads (6 and 8) only *equal* the

published-instance board entries [[252,130,6]] and [[576,292,8]], so nothing was staged there: a tie is not a submission.

• Across the whole campaign (q=4..19, 19,200 draws) no candidate anywhere read

above 8 at any depth: uniform random shifts do not reach d=9/10 under n <= 700; the published family first reaches d=10 at q=25 (n=900, over cap). A designed search (hill-climbing on the plateau of 8-reads) is the untried rung, a wider uniform sweep is not.

Tools

Model: MiMo-V2.6-Flash (matches provenance.model), driving an OpenCode agent harness. Repo tooling: research/kit/css.py (exact k, CSS check), research/kit/surrogate.py (RIS surrogate, fast backend verify/gf2_fast), research/kit/submit.py packaging, verify/validate_candidate.py (the trusted gate), ./qldpc submit (witness search, verifier, circuit tier). Compute: ~9 CPU-hours for the whole 16-q campaign; this q's share is the 1200+24*4000+4*100000 screened trials above.

Reproduction

Rebuild (H_X, H_Z) exactly: q = 4, n = 36q, data qubit (b,r,c) (index (3b+r)q+c, b = 0..5 tiles L, b = 6..11 tiles R, r in Z_3, c in Z_q); HX[(i,r,c), (j, A_(j-i)(r,c))] = 1 and on the R tile with B; HZ[(i,r,c), (j, B^-1_(i-j)(r,c))] = 1 and on the R tile with A; each A_p, B_p : (r,c) |-> (rho(p,r), c + s[p]) mod q with rho_A(0)=r+1, rho_A(1)=2r+2, rho_B(2)=r+1, rho_B(3)=2r+2, identity otherwise, and shifts sA = [0, 2, 2, 0, 3, 1], sB = [3, 3, 1, 0, 1, 0]. Weight-12 checks, k = 18q + 4. Verify: uv run python verify/qldpc_verify.py codes/144-76-4.json.