The ArcGIS server Geodatabase is 10.4.1 with Oracle 220.127.116.11. On client side, I have ArcGIS desktop 10.4.1 on Windows 10 (64bit). what Oracle client version is required? 32 bit or 64 bit? Can I use Oracle 12c client or has to be 11g 32 bit?
You'll need the 32 bit version of the Oracle client as ArcMap is a 32 bit version of the database. On the Server machine, you'll need the 64 bit version as Server is 64 bit.
Someone may comment with more knowledge but I don't believe Oracle clients are forward compatible, (using a 11g client to connect to a 12.x database), but they are backwards compatible, (using a 12c client to connect to a 11.x database). I may be wrong.