I am about to build a new house with a ham shack incorporated in it. I am considering installing number 4 copper wire in the foundation around the perimeter as my ground ring. Is it necessary to install ground rods every 16 to 20 feet around the perimeter also?
I'm going to make the assumption that you live in the United States. With that assumption, I highly recommend you start with the National Electric Code (NEC) that applies to your area. Different jurisdictions have adopted different years of the NEC as their standard. You can access the NEC Online For Free. Your local library may have a copy you can use as well.
In addition to the NEC, Tim Kuhlman, KD7RUS has a good summary of the sections of the code that apply to amateur radio.
To specifically address the Ufer connection; the 2017 version of the NEC covers the requirements for concrete-encased electrode (Ufer). It does not mention a ground rod requirement for the ground ring.