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[[136,35,9]] d ≤
n
136
k
35
d
9
kd²/n
20.846
w
12
X/Z
1.33

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Distance

X/Z asymmetry 1.33 · d_X ≤ 12, d_Z ≤ 9 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[3, 33, 50, 77, 82, 86, 93, 102, 103, 114, 119, 132]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[16, 34, 44, 49, 53, 64, 77, 115, 124]
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 3–5 (mean 4.412) · H_Z 4–5 (mean 4.5)
trapping sets H_X (1,3)×6 (2,3)×8 (3,3)×13 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 6 (1,4): 68 (1,5): 62 (2,3): 8 (2,4): 114 (2,5): 262 (2,6): 742 (2,7): 1224 (2,8): 427 (3,3): 13 (3,4): 215 (3,5): 840 (3,6): 2920 (3,7): 8371 (3,8): 17335 (3,9): 26081 (3,10): 18992 (3,11): 5085 (3,12): 1200 (3,13): 154
trapping sets H_Z (1,4)×68 (2,4)×102 (3,4)×170 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 68 (1,5): 68 (2,4): 102 (2,5): 204 (2,6): 646 (2,7): 1360 (2,8): 510 (3,4): 170 (3,5): 612 (3,6): 2244 (3,7): 7106 (3,8): 15198 (3,9): 27574 (3,10): 22848 (3,11): 6188 (3,12): 1496 (3,13): 204

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Check deletion from the board entry [[136,34,12]] (codes/136-34-12.json): row 0 of H_X removed. Dropping one stabiliser generator frees one logical qubit and cannot raise the distance.
model Claude Claude Opus 5 (claimed, not verified)
date 2026-09-21
notes Found by sweeping every single-row deletion of every small board entry and keeping those that land undominated. The parent's remaining checks are unchanged.
family check-deletion (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

[[136,35,9]] by check deletion from 136-34-12

Hypothesis

Every stabiliser generator costs one logical qubit, so deleting a single row of H_X returns a code with k + 1 logical qubits on the same n physical qubits at a distance that can only fall or stay put. The submittable cases are the ones where the drop is small enough that the result still lands undominated.

Construction

Parent: the board entry codes/136-34-12.json. Row 0 of H_X is removed and every other check is untouched.

What was swept

All 480 single-row deletions of the board entries small enough to re-derive the distance cheaply, each re-measured for k, distance and Tanner connectivity. Deletions that dropped the distance too far, or disconnected the Tanner graph, were discarded. The survivors were then screened against the live board; this is one of the few still undominated.

Distance

Two-sided random information-set search first. That search returns the minimum over both sides, so when one side is much lighter the other never surfaces even though the schema needs a witness for it. The missing side is then searched on its own in the quotient ker(H_opposite) / rowspace(H_own), by randomly permuting and reducing a kernel basis and keeping the lightest row that stays outside the row space.

Deleting a row also leaves H_X and H_Z with different shapes, so the argument order of a two-sided search stops being a reliable side label. Every witness here is classified by the conditions it actually satisfies rather than by which call produced it.

Boundary

Distance is an upper bound from witness search, not a proof. The parent's own distance claim is inherited context, not re-derived. Literature novelty is unverified.

Parity checks

X-checks 50 (max weight 12) · Z-checks 51 (max weight 12)
H_X (50 checks, sparse supports)
[3, 19, 34, 38, 64, 65, 84, 90, 91, 104, 117, 119] [4, 20, 35, 39, 65, 66, 68, 91, 92, 105, 118, 120] [5, 21, 36, 40, 66, 67, 69, 92, 93, 102, 106, 121] [6, 22, 37, 41, 51, 67, 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, 100, 101, 110, 114, 129] [14, 30, 45, 49, 58, 59, 78, 85, 101, 111, 115, 130] [15, 31, 46, 50, 59, 60, 79, 85, 86, 112, 116, 131] [16, 32, 34, 47, 60, 61, 80, 86, 87, 113, 117, 132] [0, 33, 35, 48, 61, 62, 81, 87, 88, 114, 118, 133] [1, 17, 36, 49, 62, 63, 82, 88, 89, 102, 115, 134] [12, 13, 19, 35, 54, 67, 84, 100, 106, 107, 120, 133] [13, 14, 20, 36, 51, 55, 68, 101, 107, 108, 121, 134] [14, 15, 21, 37, 52, 56, 69, 85, 108, 109, 122, 135] [15, 16, 22, 38, 53, 57, 70, 86, 109, 110, 119, 123] [0, 16, 23, 39, 54, 58, 71, 87, 110, 111, 120, 124] [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, 117, 118, 127, 131] [7, 8, 31, 47, 62, 66, 79, 95, 102, 118, 128, 132] [8, 9, 32, 48, 63, 67, 80, 96, 102, 103, 129, 133] [9, 10, 33, 49, 51, 64, 81, 97, 103, 104, 130, 134] [10, 11, 17, 50, 52, 65, 82, 98, 104, 105, 131, 135] [11, 12, 18, 34, 53, 66, 83, 99, 105, 106, 119, 132] [3, 16, 29, 30, 36, 52, 69, 82, 101, 117, 123, 124] [0, 4, 30, 31, 37, 53, 70, 83, 85, 118, 124, 125] [1, 5, 31, 32, 38, 54, 71, 84, 86, 102, 125, 126] [2, 6, 32, 33, 39, 55, 68, 72, 87, 103, 126, 127] [3, 7, 17, 33, 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, 134, 135] [11, 15, 24, 25, 48, 64, 77, 81, 96, 112, 119, 135] [12, 16, 25, 26, 49, 65, 78, 82, 97, 113, 119, 120] [0, 13, 26, 27, 50, 66, 79, 83, 98, 114, 120, 121] [1, 14, 27, 28, 34, 67, 80, 84, 99, 115, 121, 122] [2, 15, 28, 29, 35, 51, 68, 81, 100, 116, 122, 123]
H_Z (51 checks, sparse supports)
[2, 18, 37, 50, 63, 64, 83, 89, 90, 103, 116, 135] [3, 19, 34, 38, 64, 65, 84, 90, 91, 104, 117, 119] [4, 20, 35, 39, 65, 66, 68, 91, 92, 105, 118, 120] [5, 21, 36, 40, 66, 67, 69, 92, 93, 102, 106, 121] [6, 22, 37, 41, 51, 67, 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, 100, 101, 110, 114, 129] [14, 30, 45, 49, 58, 59, 78, 85, 101, 111, 115, 130] [15, 31, 46, 50, 59, 60, 79, 85, 86, 112, 116, 131] [16, 32, 34, 47, 60, 61, 80, 86, 87, 113, 117, 132] [0, 33, 35, 48, 61, 62, 81, 87, 88, 114, 118, 133] [1, 17, 36, 49, 62, 63, 82, 88, 89, 102, 115, 134] [12, 13, 19, 35, 54, 67, 84, 100, 106, 107, 120, 133] [13, 14, 20, 36, 51, 55, 68, 101, 107, 108, 121, 134] [14, 15, 21, 37, 52, 56, 69, 85, 108, 109, 122, 135] [15, 16, 22, 38, 53, 57, 70, 86, 109, 110, 119, 123] [0, 16, 23, 39, 54, 58, 71, 87, 110, 111, 120, 124] [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, 117, 118, 127, 131] [7, 8, 31, 47, 62, 66, 79, 95, 102, 118, 128, 132] [8, 9, 32, 48, 63, 67, 80, 96, 102, 103, 129, 133] [9, 10, 33, 49, 51, 64, 81, 97, 103, 104, 130, 134] [10, 11, 17, 50, 52, 65, 82, 98, 104, 105, 131, 135] [11, 12, 18, 34, 53, 66, 83, 99, 105, 106, 119, 132] [3, 16, 29, 30, 36, 52, 69, 82, 101, 117, 123, 124] [0, 4, 30, 31, 37, 53, 70, 83, 85, 118, 124, 125] [1, 5, 31, 32, 38, 54, 71, 84, 86, 102, 125, 126] [2, 6, 32, 33, 39, 55, 68, 72, 87, 103, 126, 127] [3, 7, 17, 33, 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, 134, 135] [11, 15, 24, 25, 48, 64, 77, 81, 96, 112, 119, 135] [12, 16, 25, 26, 49, 65, 78, 82, 97, 113, 119, 120] [0, 13, 26, 27, 50, 66, 79, 83, 98, 114, 120, 121] [1, 14, 27, 28, 34, 67, 80, 84, 99, 115, 121, 122] [2, 15, 28, 29, 35, 51, 68, 81, 100, 116, 122, 123]
Code ID 136-35-9 · download JSON · raw on GitHub