  Consider a stabilizer state on $n$ qudits, each of dimension $D$ with $D$
being a prime or a squarefree integer, divided into three mutually disjoint
sets or parts. Generalizing a result of Bravyi et al. [J. Math. Phys.
\textbf{47}, 062106 (2006)] for qubits (D=2), we show that up to local
unitaries on the three parts the state can be written as a tensor product of
unentangled single-qudit states, maximally entangled EPR pairs, and tripartite
GHZ states. We employ this result to obtain a complete characterization of the
properties of a class of channels associated with stabilizer error-correcting
codes, along with their complementary channels.
