Nonexistence of a projective plane of order 10

Remarks

The Bruck–Ryser–Chowla theorem does not exclude order $10$: since $10 \equiv 2 \pmod 4$, it requires $10$ to be a sum of two squares, and $10 = 1^2 + 3^2$. The proof uses coding theory. The incidence vectors of the lines of a putative plane of order $10$ span a binary linear code of length $111$. Its weight enumerator is determined by the numbers of codewords of a few small weights. A series of large computer searches showed that these numbers lead to a contradiction. The final search was completed by Clement Lam, Larry Thiel and Stanley Swiercz, and announced in 1989.

As a consequence, there is no set of $9$ mutually orthogonal Latin squares of order $10$, since such a set would give a plane of order $10$. The proof depends on computer calculations, which have since been checked independently by other computations.

Going deeper

For a student

Order $10$ is the smallest order that is not a prime power and is not excluded by the Bruck–Ryser–Chowla theorem. Orders $2, 3, 4, 5, 7, 8, 9$ are prime powers and have planes, and order $6$ is excluded by Bruck–Ryser–Chowla. The next open case is order $12$. The result is a famous early example of a theorem whose proof rests on a massive computation.