New necessary conditions on (negative) Latin square type partial difference sets in abelian groups
Partial difference sets (for short, PDSs) with parameters (n2, r(n −∈), ∈n + r2 − 3∈r, r2 − ∈r) are called Latin square type (respectively negative Latin square type) PDSs if ∈= 1 (respectively ∈= −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−1 /2 , p ≡ 1 (mod 4) 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 .