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:

  1. $N(n)\leq n-1$ for all $n\geq 2$, with equality if $n$ is a prime power. This is well known.

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

  3. $N(10)\leq 8$. This was done using a computer search. (Source)

  4. 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 $N(n)\geq n^{1/14.8}$ if $n$ is large enough, and several results of the form "If $n\geq n_\nu$ then $N(n)\geq \nu$"), but neither I nor anyone I know can add to this list, and I haven't had much luck on Google either.


1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.