[[73,46,6]] · QEC Challenge

[[73,46,6]] d ≤stabilizer

kd²/n

22.685

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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 6 · witness Pauli weight 6 (claimed upper_bound)

witness operator (Pauli string, 6 qubits)

IIIIIIIIIIIIIIIIIIXIIIIXIIIXIIIIIIIIIIXIIIIIIIIIIIXIIIIIIIIIIIIIIXIIIIIII X: [18, 23, 27, 38, 50, 65] Z: []

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 24–28 (mean 25.333)

qubit degrees S 4–12 (mean 9.37)

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

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

(1,4): 1 (1,5): 2 (1,6): 3 (1,7): 4 (1,8): 10 (1,9): 17 (1,10): 12 (1,11): 17 (1,12): 7 (2,4): 1 (2,5): 1 (2,6): 17 (2,7): 49 (2,8): 82 (2,9): 169 (2,10): 260 (2,11): 373 (2,12): 408 (2,13): 395 (2,14): 348 (2,15): 225 (2,16): 137 (2,17): 67 (2,18): 27 (2,19): 11 (2,20): 3 (3,3): 1 (3,4): 7 (3,5): 51 (3,6): 164 (3,7): 549 (3,8): 1291 (3,9): 2532 (3,10): 4385 (3,11): 6578 (3,12): 8646 (3,13): 9437 (3,14): 9028 (3,15): 7440 (3,16): 5499 (3,17): 3427 (3,18): 1759 (3,19): 779 (3,20): 303 (3,21): 103 (3,22): 28 (3,23): 8

Construction & provenance

authors @hirobumii

provenance submitted through the challenge

novelty novelty not audited

construction In the literal common rank18 check space H of the public73/37/6 CSS parent, impose nine recorded coupled coefficient constraints. Select the recorded rank27 basis from permitted weight24/28 X/Y/Z words. The discovery used a partial cyclic graph followed by five hyperplanes; this source proof directly reconstructs the final subgroup from H and its nine literal constraints.

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

builds on https://qecdb.org/codes/672e147545661c824c3fba8b, https://arxiv.org/abs/quant-ph/9910060, https://arxiv.org/abs/quant-ph/9802061

date 2026-10-18

notes Displayed presentation: maximum check weight 28, maximum qubit degree 12, 27 checks and total incidence 684.

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

[[73,46,6]] - general stabilizer subgroup code

Direction & hypothesis

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

What was searched

In the literal common rank18 check space H of the public73/37/6 CSS parent, impose nine recorded coupled coefficient constraints. Select the recorded rank27 basis from permitted weight24/28 X/Y/Z words.

The displayed presentation has maximum check weight 28, maximum qubit degree 12, 27 checks and total incidence 684.

Evidence trail

The JSON contains weight-6 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/73-46-6.json. For each checks.S row, set A[q]=1 for q in X and B[q]=1 for q in Z, using 73 columns in each half. Form S=(A|B); a column present in both halves carries Y. Preserve row and qubit order.

Coupled coefficient parity constraints: [[116112,233505],[195620,61474],[113502,28708],[242589,147496],[16480,8240],[135356,181568],[201021,148864],[212402,221696],[137435,145408]].

Public source: 73-37-6, contributed by @mathysrennela.

Construction references:

https://qecdb.org/codes/672e147545661c824c3fba8b
https://arxiv.org/abs/quant-ph/9910060
https://arxiv.org/abs/quant-ph/9802061

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

Stabilizer generators

generators 27 (max weight 28; 9 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 (27, Pauli strings on 73 qubits)

XIIXIIIIXXIIXIIIIIXIIIIIXIXIIXIXXIIIIIIXXIXIIXIXIIXIIXIIXXXIIIIIXIIIIIXXI IIIIZZZIIIZIZIIIZIIIZIZZIIZZIZIIZIIIZIIZIZIIIZZZIZIIIIIIIZIIIIIIZIZIIIIZI IXIIXXIIIIIXIIIXXIIIIIXXIIIIXIIIXXIXXXIIIIIIIXXIIIIIIIXXIXIXXIIIIXXIIIIXI IIIYIIIIYIYIIIYIYIYIIYIIYIIIIYIIIYIYYIIIYIYYIIYIYIYIIIYIYIIYIIIIIIYIYIYII IZZIIIZZIZIIZZIIZZIZIIIIZIIIIIZZZIZZZIIIIIIIIIIIIZIIIZZIZIIZZIIIIIIIIIZII IIIIIIIXXIIIIIIXIIXIIIXXIXIIXIIXIIIIIIXXIIXIIXIIIIIIXIIIXIXIXXXXXXIIIXIXI XIXIIXXIIIIIIIIIXIIIXXIIIXXIXIIXXIIXXIXIIIIXIIIIIXXXIXXXIIIIIIIIIIIIXIIIX IIYIIIIIIYIYIYIIIIIYIIIIYIYIIIYIYIYIIYIIYIIIIYIIIYIYYIIIYIYYIIYIYIYIIIYIY IIIIIIZZIIIIIIZIIZIIIZZIZIIZIIZIIIIIIZZIIZIIZIIIIIIZIIIZIZIZZZZZZIIIZIZII IYIIYYIIYYIYIIIIYIIIIIYYYIYYYIIIIIIIIIIIIYIIIYYIYIIYYIIIIIIIIIYIIIYYIIIYY IIIZZIZIIZIIZIIIIIIZZIIZIIZIIIIIIZIIIZIZIZZZZZZIIIZIZIIIIIIIIZZIIIIIIZIIZ IIXIXIIXIXIIIXIIIIXXXIIIXIIIIXXXIIXIIXIIXIIIIIXIIXIIXIXIXIIIIXIIIXIIXIIXI IIIIIIIIIIZIIIZZIZIIZZIIIIIIIIIZIIIZZIIIZZIZIIZZIIZZIZIIIIZIIIIIZZZIZZZII IIIXIIIIIXIXIIXIXXIIIIIIXXIXIIXIXIIXIIXIIXXXIIIIIXIIIIIXXIXIIXIIIIXXIIXII IZIZIIIIIZIIIIZIZIIIZIZIZIIZIIZIIIIZIIIZIZZIIIZIZZIIZIZIZIIIZIZIIZIIIIIIZ ZIIIIIZZZIIIZIZIIIZIIIZIZZIIZZIZIIZIIIZIIZIZIIIZZZIZIIIIIIIZIIIIIIZIZIIII IIIIIZZZZIIZZIIIIIZIZZIIIIIIIIIIIZIZIZIIZZIZIZIIZIZIIIIIIIZIZIIIZIIIIIZZZ YYIYIIYYIIYYIYIIIIYIIIIIYYYIYYYIIIIIIIIIIIIYIIIYYIYIIYYIIIIIIIIIYIIIYYIII ZIZIIZIIZIIIIIIIIIIIIIZZIIIZIIIIIIIIZIZIIIZZIIIIIIIZZZZZIZZIZZIZZZZZZIZIZ YIIYIIIYIIIIYYIYIYIIYYIIYIIYIYYYIYIIIIYIIIYIIYIIIYIIIIYIYYIYYIYYIIYIIIIIY IYIIIYYIIIIIIYIYIYYIIIIYIYYIIYIIIIIIIIYYYYYIYIYYIIYYIIYYYIIIIIIYIIIYIIIYY IIXIIIXXIXIIIXXIIIIIIXIXIXXIIIIXIXXIIXIIIIIIIIXXXXXIXIXXIIXXIIXXXIIIIIIXI IIYIIYIIYIIIIYIYIIYYIIYYIIIIYYYIIYIYYIYIIIIYIIIYYIYIYIIYIYYIIIYIYIIIYYIII YYYYYIIIIYIIIYYYYYYYYIYIIYIYIIIIIIIYIIYIIIIIIYIYYIIYIIIIYIIYIIIYIYIIIIIYI IIXIIIXIIIIIXXXXXIXIXXIIIIIIIXXIXIIIXXXIIXXIXIIIIXXIIXIXIXIIIXIXIXIIIXIII IYYYIIIIIIYIIIYIIIYYIYIIIYYIIIIIIYIYIYYIIIIYIYYIIYIIIIIIIIYYYYYIYIYYIIYYI XXIIIXIIXXXXIXXIIIIXXXXXIIXXXXIIXIIIIIIIXIIIIIIXXIIIXIXIXIIIIXIIIIIIIXXII

symplectic rows (A | B) (27, sparse supports)

X: [0, 3, 8, 9, 12, 18, 24, 26, 29, 31, 32, 39, 40, 42, 45, 47, 50, 53, 56, 57, 58, 64, 70, 71] Z: [] X: [] Z: [4, 5, 6, 10, 12, 16, 20, 22, 23, 26, 27, 29, 32, 36, 39, 41, 45, 46, 47, 49, 57, 64, 66, 71] X: [1, 4, 5, 11, 15, 16, 22, 23, 28, 32, 33, 35, 36, 37, 45, 46, 54, 55, 57, 59, 60, 65, 66, 71] Z: [] X: [3, 8, 10, 14, 16, 18, 21, 24, 29, 33, 35, 36, 40, 42, 43, 46, 48, 50, 54, 56, 59, 66, 68, 70] Z: [3, 8, 10, 14, 16, 18, 21, 24, 29, 33, 35, 36, 40, 42, 43, 46, 48, 50, 54, 56, 59, 66, 68, 70] X: [] Z: [1, 2, 6, 7, 9, 12, 13, 16, 17, 19, 24, 30, 31, 32, 34, 35, 36, 49, 53, 54, 56, 59, 60, 70] X: [7, 8, 15, 18, 22, 23, 25, 28, 31, 38, 39, 42, 45, 52, 56, 58, 60, 61, 62, 63, 64, 65, 69, 71] Z: [] X: [0, 2, 5, 6, 16, 20, 21, 25, 26, 28, 31, 32, 35, 36, 38, 43, 49, 50, 51, 53, 54, 55, 68, 72] Z: [] X: [2, 9, 11, 13, 19, 24, 26, 30, 32, 34, 37, 40, 45, 49, 51, 52, 56, 58, 59, 62, 64, 66, 70, 72] Z: [2, 9, 11, 13, 19, 24, 26, 30, 32, 34, 37, 40, 45, 49, 51, 52, 56, 58, 59, 62, 64, 66, 70, 72] X: [] Z: [6, 7, 14, 17, 21, 22, 24, 27, 30, 37, 38, 41, 44, 51, 55, 57, 59, 60, 61, 62, 63, 64, 68, 70] X: [1, 4, 5, 8, 9, 11, 16, 22, 23, 24, 26, 27, 28, 41, 45, 46, 48, 51, 52, 62, 66, 67, 71, 72] Z: [1, 4, 5, 8, 9, 11, 16, 22, 23, 24, 26, 27, 28, 41, 45, 46, 48, 51, 52, 62, 66, 67, 71, 72] X: [] Z: [3, 4, 6, 9, 12, 19, 20, 23, 26, 33, 37, 39, 41, 42, 43, 44, 45, 46, 50, 52, 61, 62, 69, 72] X: [2, 4, 7, 9, 13, 18, 19, 20, 24, 29, 30, 31, 34, 37, 40, 46, 49, 52, 54, 56, 61, 65, 68, 71] Z: [] X: [] Z: [10, 14, 15, 17, 20, 21, 31, 35, 36, 40, 41, 43, 46, 47, 50, 51, 53, 58, 64, 65, 66, 68, 69, 70] X: [3, 9, 11, 14, 16, 17, 24, 25, 27, 30, 32, 35, 38, 41, 42, 43, 49, 55, 56, 58, 61, 66, 67, 70] Z: [] X: [] Z: [1, 3, 9, 14, 16, 20, 22, 24, 27, 30, 35, 39, 41, 42, 46, 48, 49, 52, 54, 56, 60, 62, 65, 72] X: [] Z: [0, 6, 7, 8, 12, 14, 18, 22, 24, 25, 28, 29, 31, 34, 38, 41, 43, 47, 48, 49, 51, 59, 66, 68] X: [] Z: [5, 6, 7, 8, 11, 12, 18, 20, 21, 33, 35, 37, 40, 41, 43, 45, 48, 50, 58, 60, 64, 70, 71, 72] X: [0, 1, 3, 6, 7, 10, 11, 13, 18, 24, 25, 26, 28, 29, 30, 43, 47, 48, 50, 53, 54, 64, 68, 69] Z: [0, 1, 3, 6, 7, 10, 11, 13, 18, 24, 25, 26, 28, 29, 30, 43, 47, 48, 50, 53, 54, 64, 68, 69] X: [] Z: [0, 2, 5, 8, 22, 23, 27, 36, 38, 42, 43, 51, 52, 53, 54, 55, 57, 58, 60, 61, 63, 64, 65, 66, 67, 68, 70, 72] X: [0, 3, 7, 12, 13, 15, 17, 20, 21, 24, 27, 29, 30, 31, 33, 38, 42, 45, 49, 54, 56, 57, 59, 60, 62, 63, 66, 72] Z: [0, 3, 7, 12, 13, 15, 17, 20, 21, 24, 27, 29, 30, 31, 33, 38, 42, 45, 49, 54, 56, 57, 59, 60, 62, 63, 66, 72] X: [1, 5, 6, 13, 15, 17, 18, 23, 25, 26, 29, 38, 39, 40, 41, 42, 44, 46, 47, 50, 51, 54, 55, 56, 63, 67, 71, 72] Z: [1, 5, 6, 13, 15, 17, 18, 23, 25, 26, 29, 38, 39, 40, 41, 42, 44, 46, 47, 50, 51, 54, 55, 56, 63, 67, 71, 72] X: [2, 6, 7, 9, 13, 14, 21, 23, 25, 26, 31, 33, 34, 37, 46, 47, 48, 49, 50, 52, 54, 55, 58, 59, 62, 63, 64, 71] Z: [] X: [2, 5, 8, 13, 15, 18, 19, 22, 23, 28, 29, 30, 33, 35, 36, 38, 43, 47, 48, 50, 52, 55, 57, 58, 62, 64, 68, 69] Z: [2, 5, 8, 13, 15, 18, 19, 22, 23, 28, 29, 30, 33, 35, 36, 38, 43, 47, 48, 50, 52, 55, 57, 58, 62, 64, 68, 69] X: [0, 1, 2, 3, 4, 9, 13, 14, 15, 16, 17, 18, 19, 20, 22, 25, 27, 35, 38, 45, 47, 48, 51, 56, 59, 63, 65, 71] Z: [0, 1, 2, 3, 4, 9, 13, 14, 15, 16, 17, 18, 19, 20, 22, 25, 27, 35, 38, 45, 47, 48, 51, 56, 59, 63, 65, 71] X: [2, 6, 12, 13, 14, 15, 16, 18, 20, 21, 29, 30, 32, 36, 37, 38, 41, 42, 44, 49, 50, 53, 55, 57, 61, 63, 65, 69] Z: [] X: [1, 2, 3, 10, 14, 18, 19, 21, 25, 26, 33, 35, 37, 38, 43, 45, 46, 49, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71] Z: [1, 2, 3, 10, 14, 18, 19, 21, 25, 26, 33, 35, 37, 38, 43, 45, 46, 49, 58, 59, 60, 61, 62, 64, 66, 67, 70, 71] X: [0, 1, 5, 8, 9, 10, 11, 13, 14, 19, 20, 21, 22, 23, 26, 27, 28, 29, 32, 40, 47, 48, 52, 54, 56, 61, 69, 70] Z: []

Code ID 73-46-6 · download JSON · raw on GitHub