[[143,78,3]] · QEC Challenge

[[143,78,3]] d ≤

143

kd²/n

4.90909

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (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, 3 qubits)

[31, 34, 116]

d_Z 3 · witness weight 3 (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, 3 qubits)

[55, 79, 122]

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 1–3 (mean 2.685) · H_Z 1–3 (mean 2.769)

trapping sets H_X (1,1)×4 (2,1)×48 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 4 (1,2): 37 (1,3): 102 (2,1): 48 (2,2): 248 (2,3): 570 (2,4): 1073 (3,0): 3 (3,1): 154 (3,2): 1450 (3,3): 5529 (3,4): 10673 (3,5): 15751 (3,6): 1583 (3,7): 1813

trapping sets H_Z (1,1)×2 (2,1)×41 (3,0)×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): 29 (1,3): 112 (2,1): 41 (2,2): 219 (2,3): 480 (2,4): 1240 (3,0): 3 (3,1): 101 (3,2): 1136 (3,3): 4963 (3,4): 9596 (3,5): 18869 (3,6): 1425 (3,7): 2169

Circuit tier

syndrome-extraction memory circuits committed under circuits/143-78-3/ · canonical noise recipe, 3 rounds, stim 1.16.0

d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[1117, 1120, 14488]

d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[3822, 3870, 5621]

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/144-76-4.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

[[143,78,3]] — priced calculus E-PUNCTURE transform of codes/144-76-4.json

Direction & hypothesis

Weight-12 unrestricted CSS cell. Transform a connected CSS parent (codes/144-76-4.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/144-76-4.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/144-76-4.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/144-76-4.json by applying the E-PUNCTURE move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 32 (max weight 12) · Z-checks 33 (max weight 12)

H_X (32 checks, sparse supports)

[4, 21, 25, 35, 50, 60, 74, 86, 100, 115, 120, 131] [5, 22, 26, 36, 47, 61, 71, 83, 101, 116, 121, 132] [7, 20, 24, 38, 49, 59, 73, 85, 99, 118, 119, 134] [8, 18, 29, 39, 54, 64, 78, 90, 104, 111, 124, 135] [9, 19, 30, 40, 51, 65, 75, 87, 105, 112, 125, 136] [10, 16, 27, 41, 52, 66, 76, 88, 106, 113, 126, 137] [11, 17, 28, 42, 53, 63, 77, 89, 103, 114, 123, 138] [0, 14, 33, 43, 58, 68, 82, 94, 96, 107, 128, 139] [1, 15, 34, 44, 55, 69, 79, 91, 97, 108, 129, 140] [2, 12, 31, 45, 56, 70, 80, 92, 98, 109, 130, 141] [3, 13, 32, 46, 57, 67, 81, 93, 95, 110, 127, 142] [1, 16, 33, 37, 47, 62, 71, 86, 98, 112, 127, 132] [2, 17, 34, 38, 48, 59, 72, 83, 95, 113, 128, 133] [3, 18, 31, 35, 49, 60, 73, 84, 96, 114, 129, 134] [0, 19, 32, 36, 50, 61, 74, 85, 97, 111, 130, 131] [6, 20, 30, 42, 52, 63, 76, 87, 99, 117, 124, 137] [7, 21, 27, 39, 53, 64, 77, 88, 100, 118, 125, 138] [4, 22, 28, 40, 54, 65, 78, 89, 101, 115, 126, 135] [9, 12, 25, 45, 55, 70, 79, 94, 106, 108, 119, 140] [10, 13, 26, 46, 56, 67, 80, 91, 103, 109, 120, 141] [11, 14, 23, 43, 57, 68, 81, 92, 104, 110, 121, 142] [3, 13, 27, 45, 49, 59, 72, 83, 98, 110, 124, 139] [0, 14, 28, 46, 50, 60, 73, 84, 95, 107, 125, 140] [1, 15, 29, 43, 47, 61, 74, 85, 96, 108, 126, 141] [2, 12, 30, 44, 48, 62, 71, 86, 97, 109, 123, 142] [7, 17, 31, 41, 53, 63, 76, 87, 102, 114, 128, 135] [4, 18, 32, 42, 54, 64, 77, 88, 99, 111, 129, 136] [5, 19, 33, 39, 51, 65, 78, 89, 100, 112, 130, 137] [6, 16, 34, 40, 52, 66, 75, 90, 101, 113, 127, 138] [11, 20, 23, 37, 57, 67, 80, 91, 106, 118, 120, 131] [8, 21, 24, 38, 58, 68, 81, 92, 103, 115, 121, 132] [9, 22, 25, 35, 55, 69, 82, 93, 104, 116, 122, 133]

H_Z (33 checks, sparse supports)

[1, 12, 26, 43, 58, 60, 79, 86, 96, 107, 121, 141] [2, 13, 23, 44, 55, 61, 80, 83, 97, 108, 122, 142] [3, 14, 24, 45, 56, 62, 81, 84, 98, 109, 119, 139] [0, 15, 25, 46, 57, 59, 82, 85, 95, 110, 120, 140] [5, 16, 30, 39, 50, 64, 71, 90, 100, 111, 125, 137] [6, 17, 27, 40, 47, 65, 72, 87, 101, 112, 126, 138] [7, 18, 28, 41, 48, 66, 73, 88, 102, 113, 123, 135] [4, 19, 29, 42, 49, 63, 74, 89, 99, 114, 124, 136] [10, 20, 31, 36, 51, 69, 76, 91, 105, 116, 130, 134] [11, 21, 32, 37, 52, 70, 77, 92, 106, 117, 127, 131] [8, 22, 33, 38, 53, 67, 78, 93, 103, 118, 128, 132] [1, 13, 23, 38, 55, 70, 81, 91, 98, 108, 119, 133] [2, 14, 24, 35, 56, 67, 82, 92, 95, 109, 120, 134] [3, 15, 25, 36, 57, 68, 79, 93, 96, 110, 121, 131] [0, 12, 26, 37, 58, 69, 80, 94, 97, 107, 122, 132] [5, 17, 27, 42, 51, 62, 77, 83, 102, 112, 123, 137] [6, 18, 28, 39, 52, 59, 78, 84, 99, 113, 124, 138] [7, 19, 29, 40, 53, 60, 75, 85, 100, 114, 125, 135] [4, 16, 30, 41, 54, 61, 76, 86, 101, 111, 126, 136] [9, 20, 31, 46, 47, 66, 73, 87, 106, 116, 127, 141] [10, 21, 32, 43, 48, 63, 74, 88, 103, 117, 128, 142] [11, 22, 33, 44, 49, 64, 71, 89, 104, 118, 129, 139] [11, 13, 24, 35, 50, 67, 73, 93, 103, 110, 120, 131] [8, 14, 25, 36, 47, 68, 74, 94, 104, 107, 121, 132] [9, 15, 26, 37, 48, 69, 71, 91, 105, 108, 122, 133] [10, 12, 23, 38, 49, 70, 72, 92, 106, 109, 119, 134] [3, 17, 28, 39, 54, 63, 77, 89, 95, 114, 124, 135] [0, 18, 29, 40, 51, 64, 78, 90, 96, 111, 125, 136] [1, 19, 30, 41, 52, 65, 75, 87, 97, 112, 126, 137] [2, 16, 27, 42, 53, 66, 76, 88, 98, 113, 123, 138] [7, 20, 32, 43, 58, 59, 81, 85, 99, 118, 128, 139] [4, 21, 33, 44, 55, 60, 82, 86, 100, 115, 129, 140] [5, 22, 34, 45, 56, 61, 79, 83, 101, 116, 130, 141]

Code ID 143-78-3 · download JSON · raw on GitHub