Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line $${\mathbb{F}}_q\cup\{\infty\}$$ for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.