in my opinion to put cable under the floor could be too expensive because there is a carpet fixed to the floor.
Spur could be reasonable.
Can i feed socket from main ring which is outside the room??
you can feed the socket from the ring circuit outside the room. is this a separate circuit to the main room? not that it matters.
try lifting a small area of carpet near where the socket is to go. hopefully there will be a short floorboard to lift. possibly there will be a socket cable that is suitable to use, and with enough slack on it.
is the floor chipboard or traditional floorboards?
or
is it a new house or an old house?
lifting a few floorboards is usually much cheaper/easier/quicker/tidier than long wall chases.