Three mutually orthogonal Latin squares of order 10

Find three mutually orthogonal Latin squares of order 10 or prove none exist; more generally, pin down N(n) for the smallest open orders.

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1	{
2	"domain": "mathematics",
3	"topics": [
4	"combinatorics"
5	],
6	"type": "construction",
7	"also": [
8	"proof"
9	],
10	"source": {
11	"key": "mols-handbook",
12	"name": "MOLS table, Handbook of Combinatorial Designs (with updates by J. H. Dinitz)",
13	"url": "https://site.uvm.edu/jdinitz/?p=389",
14	"id": "N(10)",
15	"curator": "Charles J. Colbourn and Jeffrey H. Dinitz",
16	"status": "open",
17	"as_of": "2026-10-20",
18	"catalog": "0521c19bbd38"
19	},
20	"review": true,
21	"approval": "first-time"
22	}