[[93,32,7]] · QEC Challenge

[[93,32,7]] d ≤stabilizer

kd²/n

16.86

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Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them

d 7 · witness Pauli weight 7 (claimed upper_bound)

witness operator (Pauli string, 7 qubits)

IIZIIIIIZIIIIIIIIIIIIIIIIIIIIIIIZIIIIIZIZIIIIIIIIIIIZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII X: [] Z: [2, 8, 32, 38, 40, 52, 56]

certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)

check weights S 8–13 (mean 10.951)

qubit degrees S 4–11 (mean 7.183)

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

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

(1,4): 2 (1,5): 9 (1,6): 15 (1,7): 33 (1,8): 18 (1,9): 12 (1,10): 2 (1,11): 2 (2,4): 2 (2,5): 5 (2,6): 6 (2,7): 33 (2,8): 89 (2,9): 180 (2,10): 306 (2,11): 405 (2,12): 455 (2,13): 389 (2,14): 283 (2,15): 133 (2,16): 73 (2,17): 25 (2,18): 6 (3,4): 2 (3,5): 2 (3,6): 9 (3,7): 40 (3,8): 116 (3,9): 311 (3,10): 868 (3,11): 1823 (3,12): 3475 (3,13): 5716 (3,14): 8263 (3,15): 10281 (3,16): 11256 (3,17): 10580 (3,18): 8532 (3,19): 6085 (3,20): 3576 (3,21): 1872 (3,22): 749 (3,23): 261 (3,24): 70 (3,25): 15 (3,26): 1 (3,27): 1

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction Physical CNOT control89,target86 applied to literal rank61 mixed subgroup of the independent34X/30Z I635 source, followed by complete cap13 minimum-incidence rebasing. Source coefficient constraints17179869185,34359738372,8632047322. Complete source cap14 census and all214 genuine general-Pauli logical7 words give source-direct exact distance7 and resource minima. Literal vector(93,32,7,13,11,61,668).

model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)

builds on https://arxiv.org/abs/quant-ph/9705052, https://arxiv.org/abs/quant-ph/0304125

date 2026-10-18

notes Displayed presentation: maximum check weight 13, maximum qubit degree 11, 61 checks and total incidence 668.

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

[[93,32,7]] - general stabilizer subgroup code

Direction & hypothesis

Study the dimension–distance tradeoff obtained by selecting stabilizer subgroups and changing sparse presentations.

What was searched

Source coefficient constraints17179869185,34359738372,8632047322. Literal vector(93,32,7,13,11,61,668).

The displayed presentation has maximum check weight 13, maximum qubit degree 11, 61 checks and total incidence 668.

Evidence trail

The JSON contains weight-7 logical witnesses. Distance confidence: P: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/93-32-7.json. For each checks.S row, set A[q]=1 for q in X and B[q]=1 for q in Z, using 93 columns in each half. Form S=(A|B); a column present in both halves carries Y. Preserve row and qubit order.

Source coefficient constraints: ["17179869185","34359738372","8632047322"].

Physical CNOT: [89,86].

Construction references:

https://arxiv.org/abs/quant-ph/9705052
https://arxiv.org/abs/quant-ph/0304125

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Stabilizer generators

generators 61 (max weight 13; 2 mixed X/Z) (one binary symplectic matrix S = (A | B); generator i is X on A_i and Z on B_i, Y where both)

generators (61, Pauli strings on 93 qubits)

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IIIIIIIIIIIIIIIIIXIIIIIIXIIIXIIXIIIIIXIIIIIXIXIIIIIIIIIIIIIXIIIIXIIIIIIXIIIIXIIIIIIIIIIXIIIXI

symplectic rows (A | B) (61, sparse supports)

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

Code ID 93-32-7 · download JSON · raw on GitHub