An explicit family of 30 blocks meeting every 6-set of [60] in at least two points

We exhibit an explicit family $\mathcal{B}$ of $30$ subsets (``blocks'') of size $6$ of $[60]=\{1,2,\dots,60\}$ with the following property: for every $6$-subset $S\subset[60]$, there exists a block $B\in\mathcal{B}$ such that $|S\cap B|\ge 2$. The construction is fully explicit and the proof is purely combinatorial.