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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File it in the catalog (shoal-catalog 0521c19bbd38)

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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	+}