Designs Having the Parameters of \(PG_{i+1}(d+ 1,q)\)

A construction is given which uses \(\text{PG}_i(d, q)\) and \(q\) copies of \(\text{AG}_i(d, q)\) to construct designs having the parameters of \(\text{PG}_{i+1}(d+1, q)\), where \(q\) is a prime power and \(i \leq d-1\).