You might need to take his paw, and show him how to cover his business. All cats should naturally know this, but I had one once that did the same. And maybe he chooses the rug sometimes because his box isn't clean enough.
My cat doesn't cover it either but is the box covered or in a tight space? If so my cat didn't like the cover and tight space so she would go on the rug, we took the top off and took it out from under the storage because we suggest it smelt bad for them.