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

2019/05/09 by Wang, Zeying
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.03730

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. As far as we know no previous general restrictions on r were 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 (9p^4s, r(3p^2s+1),-3p^2s+r²+3r,r²+r), 1 ≤ r ≤ \frac3p^2s-12, p≡ 1 \pmod 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.

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