  Quantum error-correcting codes (QECC's) are needed to combat the inherent
noise affecting quantum processes. Using ZX calculus, we represent QECC's in a
form called a ZX diagram, consisting of a tensor network. In this paper, we
present canonical forms for CSS codes and CSS states (which are CSS codes with
0 inputs), and we show the resulting canonical forms for the toric code and
certain surface codes. Next, we introduce the notion of prime code diagrams, ZX
diagrams of codes that have a single connected component with the property that
no sequence of rewrite rules can split such a diagram into two connected
components. We also show the Fundamental Theorem of Clifford Codes, proving the
existence and uniqueness of the prime decomposition of Clifford codes. Next, we
tabulate equivalence classes of ZX diagrams under a different definition of
equivalence that allows output permutations and any local operations on the
outputs. Possible representatives of these equivalence classes are analyzed.
This work expands on previous works in exploring the canonical forms of QECC's
in their ZX diagram representations.
