← back to the board
[[97,6,11]] d ≤
n
97
k
6
d
11
kd²/n
7.48454
w
6
X/Z
1.09
g
0.0058
r
6.0
layers
2
swaps
493

Share this result

Distance

X/Z asymmetry 1.09 · d_X ≤ 12, d_Z ≤ 11 · 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 12 · witness weight 12 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 12 qubits)
[0, 7, 12, 32, 38, 45, 51, 54, 55, 56, 61, 93]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02
witness operator (support, 11 qubits)
[2, 19, 25, 33, 50, 56, 69, 72, 76, 92, 95]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 5–6 (mean 5.957)
qubit degrees H_X 1–3 (mean 2.784) · H_Z 1–3 (mean 2.825)
trapping sets H_X (1,1)×1 (2,2)×15 (3,2)×26 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 1 (1,2): 19 (1,3): 77 (2,2): 15 (2,3): 170 (2,4): 490 (3,2): 26 (3,3): 302 (3,4): 1812 (3,5): 4002 (3,6): 303 (3,7): 549
trapping sets H_Z (1,1)×2 (2,2)×12 (3,1)×1 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 2 (1,2): 13 (1,3): 82 (2,2): 12 (2,3): 124 (2,4): 544 (3,1): 1 (3,2): 18 (3,3): 233 (3,4): 1383 (3,5): 4614 (3,6): 236 (3,7): 636
witness diameter X 5.6569 · Z 7.2111 (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 @vprusso · Inherited from the laid-out parent codes/98-6-12.json: the parent's coordinates with the entry of the removed qubit (parent index 48) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04
r = 6
X checkZ checkqubit site (49)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 493 nearest-neighbor SWAPs per round in total, at most 11 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 priced calculus E-PEEL transform of codes/98-6-12.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
family other (a tag, not a ranking)
locality 2D-local bilayer (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

[[97,6,11]] — priced calculus E-PEEL transform of codes/98-6-12.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/98-6-12.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/98-6-12.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/98-6-12.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/98-6-12.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 45 (max weight 6) · Z-checks 46 (max weight 6)
H_X (45 checks, sparse supports)
[0, 1, 34, 48, 50, 95] [1, 3, 5, 49, 52, 96] [2, 4, 39, 50, 53, 63] [3, 6, 8, 51, 55, 75] [4, 7, 9, 52, 56, 69] [5, 8, 43, 53, 57, 70] [6, 10, 12, 54, 59, 81] [7, 11, 13, 48, 55, 60] [8, 12, 14, 56, 61, 76] [9, 13, 46, 57, 62, 77] [10, 15, 17, 58, 64, 86] [11, 16, 18, 49, 59, 65] [12, 17, 19, 50, 60, 66] [13, 18, 20, 61, 67, 82] [15, 21, 23, 63, 70, 90] [16, 22, 24, 51, 64, 71] [17, 23, 25, 52, 65, 72] [18, 24, 26, 53, 66, 73] [19, 25, 27, 67, 74, 87] [20, 21, 26, 68, 75, 88] [0, 21, 29, 69, 76, 93] [22, 28, 30, 54, 70, 77] [23, 29, 31, 55, 71, 78] [24, 30, 32, 56, 72, 79] [25, 31, 33, 57, 73, 80] [0, 26, 32, 74, 81, 91] [27, 28, 33, 48, 75, 92] [2, 28, 35, 58, 76, 82] [29, 34, 36, 59, 77, 83] [30, 35, 37, 60, 78, 84] [31, 36, 38, 61, 79, 85] [1, 32, 37, 62, 80, 86] [2, 33, 38, 49, 81, 94] [5, 34, 40, 64, 82, 87] [35, 39, 41, 65, 83, 88] [36, 40, 42, 66, 84, 89] [3, 37, 41, 67, 85, 90] [4, 38, 42, 51, 68, 86] [9, 39, 44, 71, 87, 91] [40, 43, 45, 72, 88, 92] [6, 41, 44, 73, 89, 93] [7, 42, 45, 54, 74, 90] [14, 43, 47, 78, 91, 94] [10, 44, 46, 79, 92, 95] [11, 45, 47, 58, 80, 93]
H_Z (46 checks, sparse supports)
[0, 7, 27, 48, 69, 74] [1, 11, 33, 48, 49, 80] [0, 2, 12, 50, 76, 81] [3, 16, 38, 49, 51, 85] [1, 4, 17, 50, 52, 86] [2, 5, 18, 49, 53, 82] [6, 22, 42, 51, 54, 89] [3, 7, 23, 52, 55, 90] [4, 8, 24, 51, 53, 56] [5, 9, 25, 52, 57, 87] [10, 28, 45, 54, 58, 92] [6, 11, 29, 55, 59, 93] [7, 12, 30, 54, 56, 60] [8, 13, 31, 55, 57, 61] [9, 14, 32, 56, 62, 91] [2, 15, 47, 58, 63, 94] [10, 16, 34, 59, 64, 95] [11, 17, 35, 58, 60, 65] [12, 18, 36, 59, 61, 66] [13, 19, 37, 60, 62, 67] [14, 20, 38, 61, 68, 94] [4, 21, 63, 68, 69] [5, 15, 22, 64, 70, 96] [16, 23, 39, 63, 65, 71] [17, 24, 40, 64, 66, 72] [18, 25, 41, 65, 67, 73] [19, 26, 42, 66, 68, 74] [3, 20, 27, 67, 75, 96] [8, 21, 28, 70, 75, 76] [9, 22, 29, 69, 71, 77] [23, 30, 43, 70, 72, 78] [24, 31, 44, 71, 73, 79] [25, 32, 45, 72, 74, 80] [6, 26, 33, 73, 75, 81] [13, 28, 34, 48, 77, 82] [14, 29, 35, 76, 78, 83] [30, 36, 46, 77, 79, 84] [31, 37, 47, 78, 80, 85] [10, 32, 38, 79, 81, 86] [19, 34, 39, 50, 83, 87] [20, 35, 40, 82, 84, 88] [36, 41, 83, 85, 89] [15, 37, 42, 84, 86, 90] [26, 39, 43, 53, 88, 91] [27, 40, 44, 87, 89, 92] [33, 43, 46, 57, 92, 94]
Code ID 97-6-11 · download JSON · raw on GitHub