As is usual, let N(n) denote the maximum size of a set of mutually orthogonal Latin squares of order n. I am wondering what results hold that bound N(n) from above; the only ones I can think of are the following:

N(n)≤n−1 for all n≥2, with equality if n is a prime power. This is well known.

N(6)=1. This is also quite famous.

N(10)≤8. This was done using a computer search. (Source)

If n=1 or 2 (mod 4), and if n is not a sum of two squares, then N(n)<n−1. This is the Bruck-Ryser Theorem from 1949, though stated in Latin squares instead of projective planes.

Are there any other results of this sort? I know of many results bounding N(n) from below (mainly Beth's result that N(n)≥n1/14.8 if n is large enough, and several results of the form "If n≥nν then N(n)≥ν"), but neither I nor anyone I know can add to this list, and I haven't had much luck on Google either.

**Answer**

Design Theory by Beth, Jungnickel & Lenz gives on page 724 the upper bounds

**Attribution***Source : Link , Question Author : Nathaniel Butler , Answer Author : Carlo Beenakker*