We consider a two-dimensional quantum memory of qubits on a torus encoding an extended Fibonacci string-net model, and construct error correction strategies when those qubits are subjected to depolarizing noise. In the case of a fixed-rate sampling noise model, we find an error correcting threshold of 4.75% with a clustering decoder. Using the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the Turaev-Viro subspace. Tensor network techniques then allow to quantitatively study the action of Pauli noise on that subspace. We perform Monte-Carlo simulations of the Fibonacci code, and compare the performance of several decoders. To the best of our knowledge, this is the first time that a threshold has been calculated for a two-dimensional error correcting code in which universal quantum computation can be performed in its code space.