  We introduce a class of models, dubbed paired twist-defect networks, that
generalize the structure of Kitaev's honeycomb model for which there is a
direct equivalence between: i) Floquet codes (FCs), ii) adiabatic loops of
gapped Hamiltonians, and iii) unitary loops or Floquet-enriched topological
orders (FETs) many-body localized phases. This formalism allows one to apply
well-characterized topological index theorems for FETs to understand the
dynamics of FCs, and to rapidly assess the code properties of many FC models.
As an application, we show that the Honeycomb Floquet code of Haah and Hastings
is governed by an irrational value of the chiral Floquet index, which implies a
topological obstruction to forming a simple, logical boundary with the same
periodicity as the bulk measurement schedule. In addition, we construct
generalizations of the Honeycomb Floquet code exhibiting arbitrary
anyon-automorphism dynamics for general types of Abelian topological order.
