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[[83,14,9]] d ≤
n
83
k
14
d
9
kd²/n
13.6627
w
8
X/Z
1.11
g
0.01
r
6.0828
layers
2
swaps
377

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Distance

X/Z asymmetry 1.11 · d_X ≤ 10, d_Z ≤ 9 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (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, 10 qubits)
[13, 21, 31, 40, 41, 51, 54, 56, 59, 73]
d_Z 9 · witness weight 9 (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, 9 qubits)
[3, 26, 34, 37, 41, 51, 53, 68, 75]
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 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 7–8 (mean 7.857)
qubit degrees H_X 1–5 (mean 3.277) · H_Z 1–5 (mean 3.313)
trapping sets H_X (1,1)×2 (2,1)×1 (3,1)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 18 (1,3): 30 (1,4): 21 (1,5): 12 (2,1): 1 (2,2): 19 (2,3): 93 (2,4): 188 (2,5): 240 (2,6): 228 (2,7): 124 (2,8): 32 (3,1): 4 (3,2): 45 (3,3): 227 (3,4): 838 (3,5): 2037 (3,6): 3208 (3,7): 3590 (3,8): 2879 (3,9): 1650 (3,10): 684 (3,11): 236 (3,12): 54 (3,13): 4
trapping sets H_Z (1,1)×2 (2,1)×1 (3,1)×3 (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): 18 (1,3): 29 (1,4): 20 (1,5): 14 (2,1): 1 (2,2): 13 (2,3): 97 (2,4): 161 (2,5): 240 (2,6): 226 (2,7): 134 (2,8): 46 (3,1): 3 (3,2): 35 (3,3): 205 (3,4): 730 (3,5): 1818 (3,6): 3024 (3,7): 3476 (3,8): 2989 (3,9): 1825 (3,10): 839 (3,11): 275 (3,12): 75 (3,13): 9
witness diameter X 5.831 · Z 7.8102 (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/84-14-10.json: the parent's coordinates with the entry of the removed qubit (parent index 15) 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 = 6.083
X checkZ checkqubit site (42)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 377 nearest-neighbor SWAPs per round in total, at most 9 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/84-14-10.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 ≤ 8 (computed)

How this code was found

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

[[83,14,9]] — priced calculus E-PEEL transform of codes/84-14-10.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (codes/84-14-10.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/84-14-10.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/84-14-10.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/84-14-10.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

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