Translations of the squares in a finite field and an infinite family of 3-designs

Abstract

Let q be a prime power with q≡−1 ( mod 4) , and let F be the finite field with q elements and Q the set of nonzero squares in F. Let G= PSL(2, q) be the special linear fractional group on Ω={∞}∪F, the projective line over F, and set V={∞}∪( Q▵( Q+1)▵( Q−1)), V=Ω⧹V , where ▵ denotes the symmetric difference. First, we consider the cardinality of intersections of some translations of Q in F and show |Q∩(Q+1)∩(Q−1)|= (q−7)/8 if 2∈Q, (q−3)/8 otherwise. Next, when 2∉ Q, we determine the structure of G V=G V , the setwise stabilizer of V or V in G, and show that the design (Ω, V G) is a 3-( q+1,( q−3)/2, λ) design, where λ= (q−3)(q−5)(q−7)/64 for p≠3, (q−3)(q−5)(q−7)/(3·64) for p=3. This is a new infinite family of 3-designs.