Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , … , p − 1 } , where p is a prime and A = { A 1 , A 2 , … , A m } is a family of subsets of [ n ] = { 1 , 2 , … , n } such that | A i | ( mod p ) ∈ K for all A i ∈ A and | A i ∩ A j | ( mod p ) ∈ L for i ≠ j . In 1991, Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n ≥ 2 s − r . In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n ≥ 2 s − 2 r + 1 or n ≥ s + max 1 ≤ i ≤ r k i , then | A | ≤ n − 1 s + n − 1 s − 1 + ⋯ + n − 1 s − 2 r + 1 . This result strengthens both the upper bound of Alon–Babai–Suzuki’s conjecture and Qian and Ray-Chaudhuri’s result, when n ≥ 2 s − 2 .