← back to the board
[[135,46,6]] d ≤
n
135
k
46
d
6
kd²/n
12.2667
w
12
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 12, 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 6 · witness weight 6 (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, 6 qubits)
[14, 31, 36, 80, 85, 102]
d_Z 6 · witness weight 6 (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, 6 qubits)
[14, 31, 36, 80, 85, 102]
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 12 · H_Z 12
qubit degrees H_X 1–5 (mean 4.0) · H_Z 1–5 (mean 3.911)
trapping sets H_X (1,1)×1 (2,2)×6 (3,2)×12 (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): 4 (1,3): 28 (1,4): 63 (1,5): 39 (2,2): 6 (2,3): 60 (2,4): 214 (2,5): 574 (2,6): 808 (2,7): 696 (2,8): 184 (3,2): 12 (3,3): 153 (3,4): 731 (3,5): 2601 (3,6): 7200 (3,7): 14260 (3,8): 18594 (3,9): 15897 (3,10): 7732 (3,11): 1941 (3,12): 416 (3,13): 41
trapping sets H_Z (1,1)×1 (2,1)×2 (3,2)×19 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 1 (1,2): 5 (1,3): 35 (1,4): 58 (1,5): 36 (2,1): 2 (2,2): 11 (2,3): 67 (2,4): 283 (2,5): 583 (2,6): 803 (2,7): 579 (2,8): 167 (3,2): 19 (3,3): 223 (3,4): 923 (3,5): 3306 (3,6): 8360 (3,7): 14691 (3,8): 17706 (3,9): 13670 (3,10): 6261 (3,11): 1674 (3,12): 318 (3,13): 45

Circuit tier

syndrome-extraction memory circuits committed under circuits/135-46-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 6 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[78166, 78171, 78236, 78238, 78247, 78262]
d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[78216, 78217, 78328, 78330, 78337, 78348]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus E-PUNCTURE transform of codes/136-37-9.json
model Beam 1.0 (claimed, not verified)
date 2026-11-02
family other (a tag, not a ranking)
locality unrestricted (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

[[135,46,6]] — priced calculus E-PUNCTURE transform of codes/136-37-9.json

Direction & hypothesis

Weight-12 unrestricted CSS cell. Transform a connected CSS parent (codes/136-37-9.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-PUNCTURE. Parent: codes/136-37-9.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/136-37-9.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/136-37-9.json by applying the E-PUNCTURE move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 45 (max weight 12) · Z-checks 44 (max weight 12)
H_X (45 checks, sparse supports)
[2, 18, 33, 37, 63, 64, 83, 89, 90, 103, 116, 118] [3, 19, 34, 38, 64, 65, 67, 90, 91, 104, 117, 119] [4, 20, 35, 39, 65, 66, 68, 91, 92, 101, 105, 120] [5, 21, 36, 40, 50, 66, 69, 92, 93, 102, 106, 121] [6, 22, 37, 41, 50, 51, 70, 93, 94, 103, 107, 122] [7, 23, 38, 42, 51, 52, 71, 94, 95, 104, 108, 123] [8, 24, 39, 43, 52, 53, 72, 95, 96, 105, 109, 124] [9, 25, 40, 44, 53, 54, 73, 96, 97, 106, 110, 125] [10, 26, 41, 45, 54, 55, 74, 97, 98, 107, 111, 126] [11, 27, 42, 46, 55, 56, 75, 98, 99, 108, 112, 127] [12, 28, 43, 47, 56, 57, 76, 99, 100, 109, 113, 128] [13, 29, 44, 48, 57, 58, 77, 84, 100, 110, 114, 129] [14, 30, 45, 49, 58, 59, 78, 84, 85, 111, 115, 130] [15, 31, 33, 46, 59, 60, 79, 85, 86, 112, 116, 131] [0, 16, 35, 48, 61, 62, 81, 87, 88, 101, 114, 133] [11, 12, 18, 34, 53, 66, 83, 99, 105, 106, 119, 132] [12, 13, 19, 35, 50, 54, 67, 100, 106, 107, 120, 133] [13, 14, 20, 36, 51, 55, 68, 84, 107, 108, 121, 134] [14, 15, 21, 37, 52, 56, 69, 85, 108, 109, 118, 122] [0, 1, 24, 40, 55, 59, 72, 88, 111, 112, 121, 125] [1, 2, 25, 41, 56, 60, 73, 89, 112, 113, 122, 126] [2, 3, 26, 42, 57, 61, 74, 90, 113, 114, 123, 127] [3, 4, 27, 43, 58, 62, 75, 91, 114, 115, 124, 128] [4, 5, 28, 44, 59, 63, 76, 92, 115, 116, 125, 129] [5, 6, 29, 45, 60, 64, 77, 93, 116, 117, 126, 130] [6, 7, 30, 46, 61, 65, 78, 94, 101, 117, 127, 131] [7, 8, 31, 47, 62, 66, 79, 95, 101, 102, 128, 132] [8, 9, 32, 48, 50, 63, 80, 96, 102, 103, 129, 133] [9, 10, 16, 49, 51, 64, 81, 97, 103, 104, 130, 134] [10, 11, 17, 33, 52, 65, 82, 98, 104, 105, 118, 131] [2, 15, 28, 29, 35, 51, 68, 81, 100, 116, 122, 123] [0, 4, 30, 31, 37, 53, 70, 83, 85, 101, 124, 125] [1, 5, 31, 32, 38, 54, 67, 71, 86, 102, 125, 126] [2, 6, 16, 32, 39, 55, 68, 72, 87, 103, 126, 127] [3, 7, 16, 17, 40, 56, 69, 73, 88, 104, 127, 128] [4, 8, 17, 18, 41, 57, 70, 74, 89, 105, 128, 129] [5, 9, 18, 19, 42, 58, 71, 75, 90, 106, 129, 130] [6, 10, 19, 20, 43, 59, 72, 76, 91, 107, 130, 131] [7, 11, 20, 21, 44, 60, 73, 77, 92, 108, 131, 132] [8, 12, 21, 22, 45, 61, 74, 78, 93, 109, 132, 133] [9, 13, 22, 23, 46, 62, 75, 79, 94, 110, 133, 134] [10, 14, 23, 24, 47, 63, 76, 80, 95, 111, 118, 134] [11, 15, 24, 25, 48, 64, 77, 81, 96, 112, 118, 119] [0, 13, 26, 27, 33, 66, 79, 83, 98, 114, 120, 121] [1, 14, 27, 28, 34, 50, 67, 80, 99, 115, 121, 122]
H_Z (44 checks, sparse supports)
[1, 17, 36, 49, 62, 63, 82, 88, 89, 102, 115, 134] [2, 18, 33, 37, 63, 64, 83, 89, 90, 103, 116, 118] [3, 19, 34, 38, 64, 65, 67, 90, 91, 104, 117, 119] [4, 20, 35, 39, 65, 66, 68, 91, 92, 101, 105, 120] [5, 21, 36, 40, 50, 66, 69, 92, 93, 102, 106, 121] [6, 22, 37, 41, 50, 51, 70, 93, 94, 103, 107, 122] [7, 23, 38, 42, 51, 52, 71, 94, 95, 104, 108, 123] [8, 24, 39, 43, 52, 53, 72, 95, 96, 105, 109, 124] [9, 25, 40, 44, 53, 54, 73, 96, 97, 106, 110, 125] [10, 26, 41, 45, 54, 55, 74, 97, 98, 107, 111, 126] [11, 27, 42, 46, 55, 56, 75, 98, 99, 108, 112, 127] [12, 28, 43, 47, 56, 57, 76, 99, 100, 109, 113, 128] [13, 29, 44, 48, 57, 58, 77, 84, 100, 110, 114, 129] [14, 30, 45, 49, 58, 59, 78, 84, 85, 111, 115, 130] [15, 31, 33, 46, 59, 60, 79, 85, 86, 112, 116, 131] [0, 16, 35, 48, 61, 62, 81, 87, 88, 101, 114, 133] [11, 12, 18, 34, 53, 66, 83, 99, 105, 106, 119, 132] [12, 13, 19, 35, 50, 54, 67, 100, 106, 107, 120, 133] [13, 14, 20, 36, 51, 55, 68, 84, 107, 108, 121, 134] [14, 15, 21, 37, 52, 56, 69, 85, 108, 109, 118, 122] [0, 1, 24, 40, 55, 59, 72, 88, 111, 112, 121, 125] [1, 2, 25, 41, 56, 60, 73, 89, 112, 113, 122, 126] [2, 3, 26, 42, 57, 61, 74, 90, 113, 114, 123, 127] [3, 4, 27, 43, 58, 62, 75, 91, 114, 115, 124, 128] [4, 5, 28, 44, 59, 63, 76, 92, 115, 116, 125, 129] [5, 6, 29, 45, 60, 64, 77, 93, 116, 117, 126, 130] [6, 7, 30, 46, 61, 65, 78, 94, 101, 117, 127, 131] [7, 8, 31, 47, 62, 66, 79, 95, 101, 102, 128, 132] [8, 9, 32, 48, 50, 63, 80, 96, 102, 103, 129, 133] [9, 10, 16, 49, 51, 64, 81, 97, 103, 104, 130, 134] [2, 15, 28, 29, 35, 51, 68, 81, 100, 116, 122, 123] [1, 5, 31, 32, 38, 54, 67, 71, 86, 102, 125, 126] [2, 6, 16, 32, 39, 55, 68, 72, 87, 103, 126, 127] [3, 7, 16, 17, 40, 56, 69, 73, 88, 104, 127, 128] [4, 8, 17, 18, 41, 57, 70, 74, 89, 105, 128, 129] [5, 9, 18, 19, 42, 58, 71, 75, 90, 106, 129, 130] [6, 10, 19, 20, 43, 59, 72, 76, 91, 107, 130, 131] [7, 11, 20, 21, 44, 60, 73, 77, 92, 108, 131, 132] [8, 12, 21, 22, 45, 61, 74, 78, 93, 109, 132, 133] [9, 13, 22, 23, 46, 62, 75, 79, 94, 110, 133, 134] [10, 14, 23, 24, 47, 63, 76, 80, 95, 111, 118, 134] [11, 15, 24, 25, 48, 64, 77, 81, 96, 112, 118, 119] [0, 13, 26, 27, 33, 66, 79, 83, 98, 114, 120, 121] [1, 14, 27, 28, 34, 50, 67, 80, 99, 115, 121, 122]
Code ID 135-46-6 · download JSON · raw on GitHub