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

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)/8if 2∈Q,(q−3)/8otherwise.Next, when 2∉Q, we determine the structure of GV=GV, the setwise stabilizer of V or V in G, and show that the design (Ω,VG) is a 3-(q+1,(q−3)/2,λ) design, whereλ=(q−3)(q−5)(q−7)/64for p≠3,(q−3)(q−5)(q−7)/(3·64)for p=3.This is a new infinite family of 3-designs.