New necessary conditions on (negative) Latin square type partial difference sets in abelian groups

Abstract

Partial difference sets (for short, PDSs) with parameters (n2, r(n−ϵ), ϵn+r2−3ϵr, r2−ϵr) are called Latin square type (respectively negative Latin square type) PDSs if ϵ=1 (respectively ϵ=−1). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of non-prime power order a2b2, where gcd⁡(a,b)=1, a>1, and b is an odd positive integer ≥3. Very few general restrictions on r were previously known. Our restrictions are particularly useful when a is much larger than b. As an application, we show that if there exists an abelian negative Latin square type PDS with parameter set (9p4s,r(3p2s+1),−3p2s+r2+3r,r2+r), 1≤r≤3p2s−12, p≡1(mod4) a prime number and s is an odd positive integer, then there are at most three possible values for r. For two of these three r values, J. Polhill gave constructions in 2009 [10].