Good Qantum LDPC Codes with Linear Time Decoders

We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain (F-2(mxm))(V) ->(delta 0) (F-2(m))(E) ->(delta 0) F-2(F) where V(X-checks) are the vertices, E (qubits) are the edges, and F (Z-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes C-A, C-B : F-2(m) -> F-2(Delta) where Delta is the regularity of the underlying Cayley graphs. One of the main ingredients in the analysis is a proof of an essentially-optimal robustness property for the tensor product of two random codes.