Seven qubits protecting a qutrit: does a ((7,3,3)) code exist?

Find a three-dimensional subspace of seven qubits that corrects any single-qubit error, or prove that no ((7,3,3)) code exists.

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1	{
2	"domain": "physics-astronomy",
3	"topics": [
4	"quantum-information",
5	"coding-theory"
6	],
7	"type": "construction",
8	"also": [
9	"proof"
10	],
11	"cover": {
12	"motif": "spheres"
13	},
14	"source": {
15	"key": "qiqcop",
16	"name": "QIQCOP Zoo",
17	"url": "https://qiqc-op.com/problem/op_4765bc6f2ac91d46/",
18	"id": "op_4765bc6f2ac91d46",
19	"curator": "Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo",
20	"poser": "Isaac Chuang, Andrew Cross, Graeme Smith, John Smolin and Bei Zeng",
21	"status": "unsolved",
22	"as_of": "2026-10-20",
23	"catalog": "0521c19bbd38"
24	},
25	"review": true,
26	"approval": "first-time"
27	}