Maximum number of edges on a uniform hypergraph

Question

I need to find the maximum number of hyperedges that can be drawn in a hypergraph, such that,

1. There are $8$ vertices.
2. Every edge contains exactly $4$ vertices.
3. Every edge should have exactly $2$ vertices in common with every other edge.

I got the answer to be $7$ by brute force. I believe there's a more elegant solution. If I were to name these criteria as the $8-4-2$ graph, then my aim is to generalize the solution to $16-8-4, 32-16-8,$ etc. Thanks in advance