New infinite families of 3-designs from algebraic curves over

In this paper, we show that the stabilizer subgroup of D f + = { a ∈ F q | f ( a ) ∈ ( F q × ) 2 } for a f ∈ F q [ x ] without multiple roots can be derived from the stabilizer of D f 0 = { a ∈ F ¯ q | f ( a ) = 0 } ∪ { ∞ } . As an application, we construct a family of 3-designs such as 3 − ( q + 1 , q − 1 2 , ( q − 1 ) ( q − 3 ) ( q − 5 ) 16 ) , where q is a prime power such that q ≡ 3 ( mod 4 ) and q ≥ 59 .