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

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Department of Mathematical Sciences


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 rof 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 bis an odd positive integer ≥3. Ve r y few general restrictions on rwere previously known. Our restrictions are particularly useful when ais 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 sis an odd positive integer, then there are at most three possible values for r. For two of these three rvalues, J. Polhill gave constructions in 2009 [10].

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Journal of Combinatorial Theory, Series A