[[38,10,3]] — check-deleted derivative of the board's [[40,10,4]]

Direction & hypothesis

Target cell: weight-6 x local-2d-bilayer at the small-n end (the gate buckets this code under weight-6; its own max check weight is 5). The board's codes/40-10-4.json (n=40, k=10, d=4, w=5, one-layer layout, interaction radius 5.3852) sits one check away from a lighter code: delete one X stabilizer and one Z stabilizer, drop the two qubits that go degree-one, and k is expected to hold at 10 while n falls to 38. The two board points at d=3 that this shape can beat are codes/42-8-3.json (n=42, w=6) and codes/45-9-3.json (n=45, w=5): same distance, larger n, smaller k.

What was searched

Bounded check deletion plus degree-one r=1 removal applied to small board codes (campaign label "bounded removal", 2026-09-17). This candidate is one survivor of that screen. Screening used small random-information-sampling budgets; the survivor was then re-screened by the repo's trusted gate at a fresh random seed, and only then packaged.

Evidence trail

Trusted gate verify/validate_candidate.py (seed 351417379): passed true, no lighter logical in 4020 RIS trials, no exact duplicate, no WL equivalent, board_advancing true, advances by k, n and w, cell weight-6 x local-2d-bilayer.

Packaging ladder: the default 20000-trial witness search per side plus a 2000000-trial accelerator pass returned d_X<=3 and d_Z<=3, so the claim is d<=3 as a witness-backed upper bound, not an exact distance. Structural verification: 14 X checks, 14 Z checks, max check weight 5, k=10, score kd^2/n = 2.368. Circuit tier verified at d_circ <= 2 (X 2, Z 2) over 3 rounds, generic sequential schedule.

Dead ends

Two closer siblings did not survive refutation: [[39,9,4]] and [[39,10,4]] were refuted by weight-3 Z logicals and are only claimable at d=3. Once this code lands both are strictly dominated by it (n=38 < 39 and k=10 >= both), so neither is submitted.

Larger candidates from the same screen collapsed under the accelerator pass: [[288,8,26]] tightened to d<=24 (that point is already on the board), [[312,8,30]] tightened to d<=23 (then strictly dominated), [[480,4,36]] tightened to d<=32 (advances no board point). [[450,8,28]] passed the gate but is dominated by the board's [[434,10,28]]. A separate SAT bound campaign over the remaining cells returned zero models (UNSAT at the probed weight) after roughly 43 CPU-hours and was stopped.

Tools

MiMo-V2.6-Flash (provenance.model) found and reconstructed the code: it identified codes/40-10-4.json as the parent, applied check deletion plus degree-one removal, and rebuilt H_X, H_Z and the inherited layout. Reproduction below was written from that reconstruction and re-verified independently. Verification used verify/validate_candidate.py, the submission CLI in cli/qldpc.py with its default 20000-trial witness search and 2000000-trial accelerator, and verify/check_prose.py. Local CPU only, 10 cores, no paid compute.

Reproduction

Rebuild H_X and H_Z from codes/40-10-4.json using zero-based indices: delete qubit 19 and qubit 31, reindex survivors to 0..37, delete X row 4 (parent row {3, 8, 19, 29, 38}) and Z row 5 (parent row {14, 18, 26, 31, 34}), then drop those two qubits from every remaining row. Keep the parent's layout columns for the surviving qubits (1 layer, interaction radius 5.385164807134504). Expected result: 14 X rows, 14 Z rows, max check weight 5, k = 38 - rank(H_X) - rank(H_Z) = 10. Pass the draft to ./qldpc submit with --authors @mathysrennela --note-file notes/38-10-3.md, keep the witnesses the CLI returns, and re-run verify/validate_candidate.py on the draft before trusting the distance. Do not replace the returned distance with a hand-computed number.