Is it known/proven what the smallest quantum error correction code is that can correct arbitrary two-qubit Pauli errors? I can think of the nested/concatenated 5-qubit code or a 25-qubit version of the Shor (repetition) code, but I am not sure if there are codes requiring fewer qubits.
Another good place to find codes with your desired parameters is this website: http://www.codetables.de/
The standard format to describe a quantum code is [[n,k,d]], where n is the number of physical qubits whose joint entangled state stores the logical information, k is the number of logical qubits encoded, and d is the distance to which they are protected.
The table is n vs k, where the distance d is the # in the box, so the smallest distance-5 code is the (11,1) entry