Classification of semiregular relative difference sets with gcd(λ, n)=1 attaining Turyn’s bound

Abstract

Suppose a (λn,n,λn,λ) relative difference set exists in an abelian group G=S×H, where |S|=λ, |H|=n2, gcd(λ,n)=1, and λ is self-conjugate modulo λn. Then λ is a square, say λ=u2, and exp(S) divides u by Turyn’s exponent bound. We classify all such relative difference sets with exp(S)=u. We also show that n must be a prime power if an abelian (λn,n,λn,λ) RDS with gcd(λ,n)=1 exists and λ is self-conjugate modulo n.