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[[36,9,4]] d =
n
36
k
9
d
4
kd²/n
4.0
w
12
X/Z
1
g
0.0059
r
5.099
layers
2
swaps
68

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 8, w_Z = 12 (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)
[13, 16, 17, 23]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[26, 29, 32, 35]
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 8 · H_Z 12
qubit degrees H_X 8 · H_Z 4
trapping sets H_X (1,8)×36 (2,8)×24 (3,8)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,8): 36 (2,8): 24 (2,10): 24 (2,12): 336 (2,14): 168 (3,8): 60 (3,10): 144 (3,12): 1152 (3,14): 816 (3,16): 3432 (3,18): 864 (3,20): 384
trapping sets H_Z (1,4)×36 (2,2)×24 (3,2)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 36 (2,2): 24 (2,4): 204 (2,6): 312 (3,2): 336 (3,4): 1572 (3,6): 3408 (3,8): 1320 (3,10): 96
witness diameter X 4.0 · Z 1.4142 (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
layout contributed by @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 5.099
X checkZ checkqubit site (18)2 qubits stacked (2 layers)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 68 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 AMC3 on Z_2 x Z_2 x Z_3; Koszul polynomials A=[(0,0,0),(0,0,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,1,0),(1,0,1)], C=[(0,0,0),(0,0,1),(1,0,0),(1,1,1)]; identity-fixed, symmetry-reduced by axis permutations and sign flips; screened with the research surrogate.
model GPT-5.6 Luna (claimed, not verified)
date 2026-08-21
family other (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight > 8 (computed)

How this code was found

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

[[36,9,4]] — AMC3 weight-4 Koszul candidate

Direction & hypothesis

Target the unrestricted weight-any cell (raw check weight 12) with a new Abelian-multicycle (AMC) construction. The AMC stage of the optional-future research plan was previously blocked by the absence of any AMC constructor in the repository; a minimal, independently specified constructor was added (Koszul boundary maps over F_2[Z_2 x Z_2 x Z_3]). The hypothesis was that a symmetry-reduced weight-3/4 AMC3 sweep over small orders (n <= 200) could surface a non-dominated code in a sparse unrestricted cell.

What was searched

A bounded, seeded, symmetry-reduced AMC3 weight-4 sweep over the grids (2,2,2), (2,2,3), (2,3,3), (2,3,5), with 400 orbits / 20,000 sampled triples / 400-trial RIS distance screen per grid, seed 20260821. Each of the three Koszul polynomials is identity-fixed with a 4-monomial support. Symmetry reduction quotients by axis permutations and independent sign flips. The quotient-lattice shortest-cycle heuristic (smallest subset of non-identity generator monomials summing to 0 mod the orders) rejects candidates with a size-<=2 relation before the distance screen (may reject, never promotes). Exact CSS / rank / k / row-weight checks run before any distance screen.

The survivor [[36,9,4]] is on Z_2 x Z_2 x Z_3 with Koszul polynomials A=[(0,0,0),(0,0,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,1,0),(1,0,1)], C=[(0,0,0),(0,0,1),(1,0,0),(1,1,1)], max check weight 12.

Evidence trail

Packaged through the research kit's submission path (both witnesses persisted) and run through the trusted verify/validate_candidate.py:

  • passed: true; not refuted; not an exact duplicate; not WL-equivalent.
  • novelty: board_advancing = true, dominated_by = [] (weight-9plus x

    • unrestricted cell).

  • Fresh-seed confirmation (seed 424242): rebuilt n=36 k=9, fresh 2000-trial

    • screen d<=4 (X-logical 4, Z-logical 4), validator passed with no lighter logical in 3940 RIS trials.

The claim is d<=4, a witness-backed upper bound, not an exact certificate.

Dead ends

The AMC3 weight-3 slice produced only dominated survivors (e.g. [[36,6,4]] dominated by [[24,6,4]]; [[90,6,6]] dominated). The AMC4 uniform weight-4 slices produced dominated finalists ([[96,12,8]], [[144,12,12]], [[216,6,24]], [[216,6,22]], [[216,12,15]]). The AMC4 (2,2,3,3) weights (3,3,4,4) slice was dead (0 survivors).

Tools

Model: GPT-5.6 Luna. Author: @mathysrennela. Construction: Abelian-multicycle (AMC3) Koszul boundary maps over F_2[Z_2 x Z_2 x Z_3]. GF(2) rank / CSS from the research kit, RIS screening from the research surrogate, packaging from the research kit, trusted gate from verify/validate_candidate.py.

Reproduction

Rebuild (H_X, H_Z) from the AMC3 Koszul construction on F_2[Z_2 x Z_2 x Z_3]: the three Koszul polynomials are A=[(0,0,0),(0,0,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,1,0),(1,0,1)], C=[(0,0,0),(0,0,1),(1,0,0),(1,1,1)], identity-fixed, symmetry-reduced by axis permutations and sign flips. The exact checks are the complete reproducible artifact in codes/36-9-4.json. The sweep used 400 orbits / 20,000 sampled triples / 400-trial RIS screen per grid, seed 20260821.

Parity checks

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