Why is the array used for a restricted hashing scheme always sized to a 4k + 3 prime?
The mentioned scheme is the most probably Quadratic probing collision resolution in the Hash Table, so it is non-perfect.

The $4k + 3$ prime size of the table is to allow all cells to be probed, the size of different length will result in non accessible cells in the array, degrading performance and wasting space.

Please read about limitations of QP.