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We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) $\overline{CCZ}$ between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which $CCZ$ should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic $X$ and measurement errors and find evidence for a threshold $p_c \sim 0.1\%$ in all three codes. We expect that this threshold could be improved by optimisation of the decoder.

Braiding defects in topological stabiliser codes can be used to fault-tolerantly implement logical operations. Twists are defects corresponding to the end-points of domain walls and are associated with symmetries of the anyon model of the code. We consider twists in multiple copies of the 2d surface code and identify necessary and sufficient conditions for considering these twists as anyons: namely that they must be self-inverse and that all charges which can be localised by the twist must be invariant under its associated symmetry. If both of these conditions are satisfied the twist and its set of localisable anyonic charges reproduce the behaviour of an anyonic model belonging to a hierarchy which generalises the Ising anyons. We show that the braiding of these twists results in either (tensor products of) the S gate or (tensor products of) the CZ gate. We also show that for any number of copies of the 2d surface code the application of H gates within a copy and CNOT gates between copies is sufficient to generate all possible twists.