The existence of simple \(S_ 3(3,4,\nu)\) (Q1823248)

Here \(\nu\) is the number of elements in a set, and \(S_ 3(3,4,\nu)\) refers to a 3-design on \(\nu\) elements, each block being of size 4, and every 3-triplet of \(\nu\) elements appears exactly in 3 blocks. It is well-known that \(S_ 3(3,4,\nu)\) exists if and only if \(\nu\) is even, but the construction methods proving this result lead to configurations with repeated blocks. In this note the authors give an elementary construction technique and establish the existence of simple \(S_ 3(3,4,\nu)s\) for all even \(\nu >4\).