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[[36,6,4]] d =
n
36
k
6
d
4
kd²/n
2.667
w
6
X/Z
1
g
0.0417
r
4.0
layers
1
swaps
65

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[4, 11, 17, 27]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[2, 5, 11, 17]
certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT
X: no logical < 4 exists; Z: no logical < 4 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–6 (mean 5.667) · H_Z 4–6 (mean 5.467)
qubit degrees H_X 1–4 (mean 2.361) · H_Z 1–4 (mean 2.278)
trapping sets H_X (1,1)×7 (2,1)×17 (3,1)×58 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 7 (1,2): 15 (1,3): 8 (1,4): 6 (2,1): 17 (2,2): 42 (2,3): 56 (2,4): 32 (2,5): 21 (2,6): 10 (3,1): 58 (3,2): 137 (3,3): 245 (3,4): 277 (3,5): 254 (3,6): 133 (3,7): 50 (3,8): 10 (3,9): 7 (3,10): 3
trapping sets H_Z (1,1)×7 (2,1)×23 (3,1)×52 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 7 (1,2): 15 (1,3): 11 (1,4): 3 (2,1): 23 (2,2): 39 (2,3): 51 (2,4): 41 (2,5): 13 (2,6): 2 (3,1): 52 (3,2): 164 (3,3): 246 (3,4): 272 (3,5): 205 (3,6): 93 (3,7): 33 (3,8): 8
witness diameter X 4.1231 · Z 3.6056 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4
X checkZ checkqubit site (36)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 65 nearest-neighbor SWAPs per round in total, at most 5 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Locality-constrained SAT discovery: qubits on a grid, each check anchored within radius 2.0, weight <= 6, t=3 detection CNF (d >= 4 by construction); CaDiCaL enumeration, RIS screening, MILP-exact distance confirmation. Per-code parameters in the construction field below and in the research note.
model GLM 5.3 Flash (claimed, not verified)
date 2026-09-06
notes Distinct construction among the [[36,5,4]] co-entries from the same campaign: different check matrix (fingerprint eb3b57b1575c), gate dedup passed. Checked against existing board entries at gate time; not equivalent to any of them. MILP-exact d=4 both sides (documented in the research note).
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[36,6,4]] — SAT-discovered weight-6 single-layer checkerboard code (a)

Direction & hypothesis

Target cell: local-2d-single × weight-6. Before this campaign the cell held 6 entries with best kd²/n = 3.56 ([[18,4,4]]); the (n, 6, 4) point was unoccupied. Hypothesis (the campaign's 2026-09-02 postmortem identified t=3 on small grids as the open move): t=3 detection — a CNF requiring every weight-≤3 Pauli error to be detected — forces d ≥ 4 by construction, and a SAT enumeration over locality-constrained anchor incidence can reach k ≥ 6 at d ≥ 4 on small grids where free-form SAT search collapses.

What was searched

Locality-constrained SAT enumeration (pysat CaDiCaL153, streamed CNF): qubits on a 6 grid, 15 generator rows per side, every check anchored within Euclidean radius 2.0 of a SAT-chosen anchor site, per-row weight ≤ 6 (sequential-counter encoding), t=3 detection clauses over all weight-≤3 errors, solutions blocked on the check-incidence pattern (one code per distinct matrix). ~50 raw solutions per configuration were screened with exact GF(2) k, an 800-trial RIS distance screen (keeping d ≥ 4 only), a board dominance pre-filter, and the trusted gate. This code is the configuration whose check matrix matches codes/36-6-4.json (fingerprint eb3b57b1575c); the full sweep parameters are in provenance.construction.

Evidence trail

  • RIS witness search (20,000 trials + accelerator pass): weight-4

    • X-side logical and weight-4 Z-side logical, so d ≤ 4 on both sides. This is the tier the submission carries: witness-backed upper bound.

  • Exact certification in staging: verify/certify.py (scipy/HiGHS MILP, one

    • no-lighter-logical proof per logical-basis row, both sides) returned d_X = 4 and d_Z = 4 exact for this matrix. A maintainer can reproduce that with `uv run python verify/certify.py codes/36-6-4.json`; the board tier upgrades only on that server-side run.

  • Trusted gate verify/validate_candidate.py: passed, board-advancing

    • (novelty unverified against the wider literature).

Dead ends

  • t=3 at 6×6 with 15 rows/side needs tens of millions of auxiliary

    • variables (13 GB measured footprint) — out of a 16 GB laptop with this encoder; 6 with 15 rows/side is the affordable cell.

  • CryptoMiniSat (XOR-native, Gaussian elimination) budgeted out with zero

    • yields on both t=3 cells where CaDiCaL yielded codes; kissat cannot enumerate at all (pysat wrapper aborts on incremental add_clause).

  • Minisat22/Minicard lack pysat time budgets; unbounded solve/block rounds

    • burned hours at zero yield before a round cap was added.

Tools

Model: GLM 5.3 Flash (matches provenance.model). Harness: an overnight locality-SAT enumerator built on python-sat with streamed clause emission, conflict budgets and a hard round cap; exact-distance verification via verify/certify.py; screening and packaging via the repo kit (research/kit/); gate via verify/validate_candidate.py. Compute: one MacBook Air (M-series), roughly two hours per configuration ladder.

Reproduction

Rebuild from the submitted checks directly: the X and Z supports in codes/36-6-4.json are the construction. To regenerate the family: enumerate locality-constrained CSS codes with the parameters in provenance.construction (6 grid, 15 rows/side, weight ≤ 6, radius 2.0, t=3 detection, CaDiCaL backend), screen for k ≥ 4 and RIS d ≥ 4, and match the fingerprint above. Verify with uv run python verify/qldpc_verify.py codes/36-6-4.json.

Parity checks

X-checks 15 (max weight 6) · Z-checks 15 (max weight 6)
H_X (15 checks, sparse supports)
[4, 5, 8, 9, 17, 22] [6, 7, 12, 18, 19, 25] [3, 4, 7, 8, 14, 21] [24, 30, 32, 33] [1, 6, 13, 15, 20, 25] [0, 12, 13, 14, 18, 24] [10, 15, 22, 27, 29, 34] [12, 25, 30, 31] [2, 3, 5, 9, 10] [19, 20, 24, 26, 28, 31] [5, 10, 11, 16, 23, 29] [16, 21, 27, 33, 34, 35] [7, 8, 10, 13, 15, 20] [8, 9, 12, 15, 16, 20] [16, 21, 23, 27, 28, 35]
H_Z (15 checks, sparse supports)
[11, 17, 22, 29] [2, 7, 8, 9, 19, 26] [14, 21, 24, 28, 33] [15, 20, 26, 32, 33, 34] [3, 9, 13, 14, 15, 22] [0, 1, 6, 18] [26, 28, 32, 33, 35] [1, 2, 5, 9, 10, 15] [18, 19, 24, 26, 30, 31] [12, 13, 18, 20, 24, 30] [6, 12, 13, 19, 20, 25] [3, 8, 10, 16, 17, 27] [2, 4, 9, 11, 16, 21] [17, 22, 23, 27, 29, 34] [0, 2, 3, 14]
Code ID 36-6-4 · download JSON · raw on GitHub