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[[84,2,9]] d =
n
84
k
2
d
9
kd²/n
1.929
w
4
X/Z
1
g
0.0046
r
6.4031
layers
1
swaps
361

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[8, 14, 21, 27, 49, 57, 63, 69, 77]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[4, 9, 18, 49, 62, 71, 75, 78, 80]
certificate exact, d = 9 · CryptoMiniSat 5.14 SAT
X: no logical < 9 exists; Z: no logical < 9 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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×84 (2,2)×252 (3,2)×756 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 84 (2,2): 252 (3,2): 756 (3,4): 168
trapping sets H_Z (1,2)×84 (2,2)×252 (3,2)×756 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 84 (2,2): 252 (3,2): 756 (3,4): 168
witness diameter X 9.2195 · Z 14.1421 (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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/84-2-9/ · canonical noise recipe, 9 rounds, stim 1.16.0
d_circ ≤ 9 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 9 · fault-set witness of 9 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 9)
[27, 29, 47, 50, 63, 65, 117, 126, 166]
d_circ^Z 9 · fault-set witness of 9 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 9)
[56, 71, 397, 405, 428, 458, 698, 745, 753]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

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 @willzeng · Tier-0/1 pipeline: spectral (Laplacian eigenvector) initialization snapped to an integer grid, then simulated annealing over site assignments (single-qubit relocations + pair swaps, max-diameter objective, 2 seeds per layer option). Search driven by Claude Fable 5. · 2026-08-20
r = 6.403
X checkZ checkqubit site (84)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 361 nearest-neighbor SWAPs per round in total, at most 6 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 Two-term multivariate bicycle on Z_7 x Z_6; A=[[0, 0], [4, 4]], B=[[0, 0], [5, 5]]; identity-normalized x/y/xy supports; screen seed=20260816, sample=4.
builds on arXiv:2406.19151
date 2026-08-16
notes Screened at 160 trials; package witness search used 1600 trials. Distance is an upper bound. LAYOUT added 2026-08-20: a 1-layer planar layout with measured interaction radius 6.4031 (integer-grid sites, min site spacing 1.0, at most 1 qubits per site), placing the code in the local-2d-bilayer class with geometric efficiency g = 0.0046. Layout credit and method are recorded in locality.contributed_by; the code, its distance claims, and its authorship are unchanged. Part of the layout-filling effort (issue #654 discussion).
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

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

[[84,2,9]] — weight-4 multivariate-bicycle finalist

Direction & hypothesis

Targeted the unrestricted × weight-4 cell with identity-normalized two-term bicycle supports drawn from the x/y/(xy) support families. The hypothesis was that sparse multivariate supports could add useful rank and distance structure while retaining maximum check weight 4.

What was searched

The campaign generated 480 candidates across Z_l × Z_m dimensions. Candidates were screened with exact CSS/rank checks and a 160-trial RIS surrogate. The finalist was packaged with 1,600 witness-search trials.

Evidence trail

The submitted code has n=84, k=2, and maximum check weight 4. The repository witness search found X and Z logicals of weight 9, so the submitted distance is the upper bound d<=9, not an exact certification. The trusted validator passed the CSS, witness, deduplication, and refutation gates and marked the unrestricted × weight-4 cell as board-advancing.

Dead ends

The campaign discarded 108 candidates below the minimum screen distance and 35 duplicate stabilizer codes. The remaining finalists were not treated as exact-distance results.

Tools

The code was generated with the repository research kit and validated by the trusted verifier. The campaign used model GPT-5.6 Luna and seed 20260816.

Reproduction

Use the bivariate-bicycle construction on Z_7 × Z_6 with A={(0,0),(4,4)} and B={(0,0),(5,5)}. Build H_X=[A|B] and H_Z=[B^T|A^T] over GF(2).

Parity checks

X-checks 42 (max weight 4) · Z-checks 42 (max weight 4)
H_X (42 checks, sparse supports)
[0, 28, 42, 77] [1, 29, 43, 72] [2, 24, 44, 73] [3, 25, 45, 74] [4, 26, 46, 75] [5, 27, 47, 76] [6, 34, 48, 83] [7, 35, 49, 78] [8, 30, 50, 79] [9, 31, 51, 80] [10, 32, 52, 81] [11, 33, 53, 82] [12, 40, 47, 54] [13, 41, 42, 55] [14, 36, 43, 56] [15, 37, 44, 57] [16, 38, 45, 58] [17, 39, 46, 59] [4, 18, 53, 60] [5, 19, 48, 61] [0, 20, 49, 62] [1, 21, 50, 63] [2, 22, 51, 64] [3, 23, 52, 65] [10, 24, 59, 66] [11, 25, 54, 67] [6, 26, 55, 68] [7, 27, 56, 69] [8, 28, 57, 70] [9, 29, 58, 71] [16, 30, 65, 72] [17, 31, 60, 73] [12, 32, 61, 74] [13, 33, 62, 75] [14, 34, 63, 76] [15, 35, 64, 77] [22, 36, 71, 78] [23, 37, 66, 79] [18, 38, 67, 80] [19, 39, 68, 81] [20, 40, 69, 82] [21, 41, 70, 83]
H_Z (42 checks, sparse supports)
[0, 13, 42, 62] [1, 14, 43, 63] [2, 15, 44, 64] [3, 16, 45, 65] [4, 17, 46, 60] [5, 12, 47, 61] [6, 19, 48, 68] [7, 20, 49, 69] [8, 21, 50, 70] [9, 22, 51, 71] [10, 23, 52, 66] [11, 18, 53, 67] [12, 25, 54, 74] [13, 26, 55, 75] [14, 27, 56, 76] [15, 28, 57, 77] [16, 29, 58, 72] [17, 24, 59, 73] [18, 31, 60, 80] [19, 32, 61, 81] [20, 33, 62, 82] [21, 34, 63, 83] [22, 35, 64, 78] [23, 30, 65, 79] [24, 37, 44, 66] [25, 38, 45, 67] [26, 39, 46, 68] [27, 40, 47, 69] [28, 41, 42, 70] [29, 36, 43, 71] [1, 30, 50, 72] [2, 31, 51, 73] [3, 32, 52, 74] [4, 33, 53, 75] [5, 34, 48, 76] [0, 35, 49, 77] [7, 36, 56, 78] [8, 37, 57, 79] [9, 38, 58, 80] [10, 39, 59, 81] [11, 40, 54, 82] [6, 41, 55, 83]
Code ID 84-2-9 · download JSON · raw on GitHub