Hi,
One simple way to do this would be with an R-2R resistor ladder. On a Uno you could run your 15 buttons from probably just two analog input pins if you use this method. There are lots of explanations online, like this instructable which shows an 8 button array (make two of those and you're set).
This explanation by John Boxall shows how to setup an R-2R resistor ladder such that it can read more than one button at a time too.
Good luck ! Geoff