stylistic:tone
Projective planes

A set of n − 1 MOLS(n) is equivalent to a finite affine plane of order n (see Nets below). As every finite affine plane is uniquely extendable to a finite projective plane of the same order, this equivalence can also be expressed in terms of the existence of these projective planes.

As mentioned above, complete sets of MOLS(n) exist if n is a prime or prime power, so projective planes of such orders exist. Finite projective planes with an order different from these, and thus complete sets of MOLS of such orders, are not known to exist.

The only general result on the non-existence of finite projective planes is the Bruck – Ryser theorem, which says that if a projective plane of order n exists and n ≡ 1 (mod 4) or n ≡ 2 (mod 4), then n must be the sum of two (integer) squares. This rules out projective planes of orders 6 and 14 for instance, but does not guarantee the existence of a plane when n satisfies the condition. In particular, n = 10 satisfies the conditions, but no projective plane of order 10 exists, as was shown by a very long computer search, which in turn implies that there do not exist nine MOLS of order 10.

No other existence results are known. As of 2020, the smallest order for which the existence of a complete set of MOLS is undetermined is 12.