[[54,9,5]] · QEC Challenge

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[[54,9,5]] d =

kd²/n

4.167

X/Z

1.2

0.0026

6.3246

layers

swaps

262

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Distance

X/Z asymmetry 1.2 · d_X = 5, d_Z = 6 · 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 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[15, 34, 36, 47, 49]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[0, 3, 6, 9, 12, 15]

certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT

X: no logical < 5 exists; Z: no logical < 6 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)×54 (2,10)×36 (3,10)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,8): 54 (2,10): 36 (2,12): 306 (2,14): 792 (3,10): 36 (3,12): 204 (3,14): 1710 (3,16): 5166 (3,18): 8670 (3,20): 5994 (3,22): 288

trapping sets H_Z (1,4)×54 (2,4)×207 (3,2)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 54 (2,4): 207 (2,6): 774 (3,2): 72 (3,4): 1440 (3,6): 7104 (3,8): 9144 (3,10): 936

witness diameter X 4.1231 · Z 3.1623 (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

X checkZ checkqubit site (28)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 262 nearest-neighbor SWAPs per round in total, at most 8 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_3 x Z_3; Koszul polynomials A=[(0,0,0),(0,1,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,2,0),(1,1,1)], C=[(0,0,0),(0,2,0),(0,2,1),(1,2,1)]; 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

[[54,9,5]] — 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_3 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 [[54,9,5]] is on Z_2 x Z_3 x Z_3 with Koszul polynomials A=[(0,0,0),(0,1,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,2,0),(1,1,1)], C=[(0,0,0),(0,2,0),(0,2,1),(1,2,1)], 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=54 k=9, fresh 2000-trial

screen d<=5 (X-logical 5, Z-logical 6), validator passed with no lighter logical in 4660 RIS trials.

The claim is d<=5, a witness-backed upper bound, not an exact certificate.

Dead ends

The AMC3 weight-3 slice produced only dominated survivors. The AMC4 uniform weight-4 slices produced dominated finalists. The AMC4 (2,2,3,3) weights (3,3,4,4) slice was dead (0 survivors). The weight-4 slice also produced dominated [[36,9,4]]-adjacent records and [[90,9,6]]-type candidates that did not advance.

Tools

Model: GPT-5.6 Luna. Author: @mathysrennela. Construction: Abelian-multicycle (AMC3) Koszul boundary maps over F_2[Z_2 x Z_3 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_3 x Z_3]: the three Koszul polynomials are A=[(0,0,0),(0,1,1),(1,0,1),(1,1,0)], B=[(0,0,0),(0,0,1),(0,2,0),(1,1,1)], C=[(0,0,0),(0,2,0),(0,2,1),(1,2,1)], identity-fixed, symmetry-reduced by axis permutations and sign flips. The exact checks are the complete reproducible artifact in codes/54-9-5.json. The sweep used 400 orbits / 20,000 sampled triples / 400-trial RIS screen per grid, seed 20260821.

Parity checks

X-checks 54 (max weight 8) · Z-checks 18 (max weight 12)

H_X (54 checks, sparse supports)

[0, 1, 6, 13, 18, 22, 28, 30] [1, 2, 7, 14, 19, 23, 29, 31] [0, 2, 8, 12, 20, 21, 27, 32] [0, 3, 4, 16, 21, 25, 31, 33] [1, 4, 5, 17, 22, 26, 32, 34] [2, 3, 5, 15, 23, 24, 30, 35] [3, 6, 7, 10, 19, 24, 27, 34] [4, 7, 8, 11, 20, 25, 28, 35] [5, 6, 8, 9, 18, 26, 29, 33] [4, 9, 10, 15, 19, 21, 27, 31] [5, 10, 11, 16, 20, 22, 28, 32] [3, 9, 11, 17, 18, 23, 29, 30] [7, 9, 12, 13, 22, 24, 30, 34] [8, 10, 13, 14, 23, 25, 31, 35] [6, 11, 12, 14, 21, 26, 32, 33] [1, 12, 15, 16, 18, 25, 28, 33] [2, 13, 16, 17, 19, 26, 29, 34] [0, 14, 15, 17, 20, 24, 27, 35] [0, 6, 7, 16, 36, 40, 46, 48] [1, 7, 8, 17, 37, 41, 47, 49] [2, 6, 8, 15, 38, 39, 45, 50] [0, 1, 3, 10, 39, 43, 49, 51] [1, 2, 4, 11, 40, 44, 50, 52] [0, 2, 5, 9, 41, 42, 48, 53] [3, 4, 6, 13, 37, 42, 45, 52] [4, 5, 7, 14, 38, 43, 46, 53] [3, 5, 8, 12, 36, 44, 47, 51] [7, 9, 15, 16, 37, 39, 45, 49] [8, 10, 16, 17, 38, 40, 46, 50] [6, 11, 15, 17, 36, 41, 47, 48] [1, 9, 10, 12, 40, 42, 48, 52] [2, 10, 11, 13, 41, 43, 49, 53] [0, 9, 11, 14, 39, 44, 50, 51] [4, 12, 13, 15, 36, 43, 46, 51] [5, 13, 14, 16, 37, 44, 47, 52] [3, 12, 14, 17, 38, 42, 45, 53] [18, 24, 25, 34, 36, 37, 42, 49] [19, 25, 26, 35, 37, 38, 43, 50] [20, 24, 26, 33, 36, 38, 44, 48] [18, 19, 21, 28, 36, 39, 40, 52] [19, 20, 22, 29, 37, 40, 41, 53] [18, 20, 23, 27, 38, 39, 41, 51] [21, 22, 24, 31, 39, 42, 43, 46] [22, 23, 25, 32, 40, 43, 44, 47] [21, 23, 26, 30, 41, 42, 44, 45] [25, 27, 33, 34, 40, 45, 46, 51] [26, 28, 34, 35, 41, 46, 47, 52] [24, 29, 33, 35, 39, 45, 47, 53] [19, 27, 28, 30, 43, 45, 48, 49] [20, 28, 29, 31, 44, 46, 49, 50] [18, 27, 29, 32, 42, 47, 48, 50] [22, 30, 31, 33, 37, 48, 51, 52] [23, 31, 32, 34, 38, 49, 52, 53] [21, 30, 32, 35, 36, 50, 51, 53]

H_Z (18 checks, sparse supports)

[0, 8, 11, 15, 18, 20, 21, 35, 36, 39, 41, 50] [1, 6, 9, 16, 18, 19, 22, 33, 37, 39, 40, 48] [2, 7, 10, 17, 19, 20, 23, 34, 38, 40, 41, 49] [2, 3, 9, 14, 21, 23, 24, 29, 39, 42, 44, 53] [0, 4, 10, 12, 21, 22, 25, 27, 40, 42, 43, 51] [1, 5, 11, 13, 22, 23, 26, 28, 41, 43, 44, 52] [5, 6, 12, 17, 18, 24, 26, 32, 36, 38, 42, 47] [3, 7, 13, 15, 19, 24, 25, 30, 36, 37, 43, 45] [4, 8, 14, 16, 20, 25, 26, 31, 37, 38, 44, 46] [2, 6, 9, 17, 26, 27, 29, 30, 41, 45, 48, 50] [0, 7, 10, 15, 24, 27, 28, 31, 39, 46, 48, 49] [1, 8, 11, 16, 25, 28, 29, 32, 40, 47, 49, 50] [0, 5, 11, 12, 20, 30, 32, 33, 44, 48, 51, 53] [1, 3, 9, 13, 18, 30, 31, 34, 42, 49, 51, 52] [2, 4, 10, 14, 19, 31, 32, 35, 43, 50, 52, 53] [3, 8, 14, 15, 23, 27, 33, 35, 38, 45, 47, 51] [4, 6, 12, 16, 21, 28, 33, 34, 36, 45, 46, 52] [5, 7, 13, 17, 22, 29, 34, 35, 37, 46, 47, 53]

Code ID 54-9-5 · download JSON · raw on GitHub