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Is there a generally accepted definition of what a color code is?

I have found two definitions that I am not able to reconciliate with each other:

  1. The error correction zoo defines color codes via homogeneous simplicial complexes as the triangulation of a single simplex and qubits are placed on simplices.

  2. In An Introduction to Topological Quantum Codes in section 5, color codes are introduced as lattices that come with vertices and faces (at least, in the 2-D case) where qubits are on the vertices and the plaquettes denote X- and Z-stabilizers.

The examples that I have come accross so far (Steane, honeycomb, 4-8-8 lattice) all seem to fit the second definition, but not the first.

Are the two definitions equivalent? If yes, why? If not, what is the "correct" / generally accepted definition?

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    $\begingroup$ Some of the color codes in arxiv.org/abs/1806.02820 , such as the ones in figure 6, don't satisfy the second definition; they have boundary stabilizers that are only one Pauli type instead of both. $\endgroup$ Commented Jun 7 at 16:47
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    $\begingroup$ Thanks for pointing this out. FWIW there's another defn based on pin codes in arxiv.org/abs/1906.11394. $\endgroup$ Commented Jul 2 at 14:28

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