Sarvate-Beam Quad Systems for \(v=6\)

A Sarvate-Beam Quad System \( SB(v, 4) \) is a set \( V \) of \( v \) elements and a collection of \( 4 \)-subsets of \( V \) such that each distinct pair of elements in \( V \) occurs \( i \) times for every \( i \) in the list \( 1, 2, \ldots, \binom{v}{2} \). In this paper, we completely enumerate all Sarvate-Beam Quad Systems for \( v = 6 \).