Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code on the body-centererd cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we

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... timate the threshold for 1D string-like and 2D sheet-like logical operators to be p^(1)_3DCC≃ 1.9% and p^(2)_3DCC≃ 27.6%. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the 4- and 6-body random coupling Ising models.