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  "body": "\n## Statement\n\nThere is no [[projective plane|DANFNN8C4Y]] of [[order|K2YVMTGTBB]] $10$.\n\n## Remarks\n\nThe [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] does not exclude order $10$: since $10 \\equiv 2 \\pmod 4$, it requires $10$ to be a sum of two squares, and $10 = 1^2 + 3^2$.\nThe proof uses coding theory.\nThe incidence vectors of the lines of a putative plane of order $10$ span a binary linear code of length $111$.\nIts weight enumerator is determined by the numbers of codewords of a few small weights.\nA series of large computer searches showed that these numbers lead to a contradiction.\nThe final search was completed by Clement Lam, Larry Thiel and Stanley Swiercz, and announced in 1989.\n\nAs a consequence, there is no set of $9$ [[mutually orthogonal Latin squares|H6TN6BADVY]] of order $10$, since such a set would give a plane of order $10$.\nThe proof depends on computer calculations, which have since been checked independently by other computations.\n\n## Explanations\n\n### L1 human\n\nA projective plane of order $10$ would have $111$ points and $111$ lines, with $11$ points on each line.\nThe obvious necessary conditions do not rule it out.\nIt took years of reasoning and computer search to show that no such plane exists.\n\n### L2 human\n\nOrder $10$ is the smallest order that is not a prime power and is not excluded by the Bruck–Ryser–Chowla theorem.\nOrders $2, 3, 4, 5, 7, 8, 9$ are prime powers and have planes, and order $6$ is excluded by Bruck–Ryser–Chowla.\nThe next open case is order $12$.\nThe result is a famous early example of a theorem whose proof rests on a massive computation.\n",
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