On the existence of small quasimultiples of affine and projective planes of arbitrary order

✍ Scribed by Dieter Jungnickel

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 798 KB
• Volume: 85
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90020-i

✦ Synopsis

Denote by a(n) and p(n), respectively, the smallest positive integers ,I and p for which an &(2, n, n') and an S,,(2, n + 1, n2 + n + 1) exist. We thus consider the problem of the existence of (nontrivial) quasimultiples of atline and projective planes of arbitrary order n. The best previously known general bounds state that a(n) s n"-*-* and p(n) =Z n"-'-', provided that there exist k mutually orthogonal Latin squares of order n; this is due to Mavron, Mullin and Rosa. We substantially improve this result by showing that both a(n) and p(n) are bounded by nz9, whenever n is sufficiently large. If n has exactly k distinct prime factors, where k L 28, both bounds can be improved to nk.

We also construct many families of values of n for which much better bounds can be given; for instance, both functions are bounded by 2n for n of the type n = pq, where p and 9 are odd prime powers with p < q <2p. Another example gives a bound of n + 2 for n of the form n = 2q, q an odd prime power. Only one such family, giving a bound of (q -1)/2 for n = q + 1, q an odd prime power, was previously obtained (by Shrikhande and Singhi). Finally, we discuss the case n = 6 (where a(6) =p(6) = 2 is known) in some detail and obtain new solutions for this case.