Hi Derek, are all the 2-primitive but not 3-transitive permutation groups explicitly known? Although it should be not hard to find them from the 2-transitive permutation groups, I wonder if there is any existing list of them?

@YuichiroFujiwara To my knowledge, the answer to the latter question is not known yet. For the former question, is the answer "yes" up to isomorphisms of designs?

@YuichiroFujiwara Thanks for your comment which helps me to edit my question. (hope it is clearer now!) What do you mean by 'isomorphic' of cyclic difference set? If $D'=cD+d$ for some $c,d\in\mathbb{Z}/v\mathbb{Z}$ and $c$ coprime to $v$ then I say $D'$ is equivalent to $D$. Do you mean just this equivalence or any more?

Thanks! I've edited my question to make it more precise.

@PeterMueller: Yes. It seems that the fewer prime factors $l$ has, the easier the situation would be. So I think one may start the question when $l=p$ is prime (the $2p$th cyclotomic polynomial is also clear).

@suv....rit :Thanks! In your experiments, what is the smallest value of $l$ such that the matrix seems to be singular?