Let $S=\{0, 1, ..., 21\}$, and $T$ is a set of 4-subsets of $S$ (call these blocks), such that every 3 elements of $S$ appear in exactly one block of $T$. This is known as a Steiner quadruple system. $$m = \min_{t \in T} \{a+b+c+d \mid (a,b,c,d) =t \in T\}.$$ What is the maximum value for $m$?
I know $m \leq 24$. So, I am trying to determine if $m$ can attain 24.