  We study a simple, self-dual, rational, and $C_2$-cofinite vertex operator
algebra of CFT-type whose simple current modules are graded by
$\mathbb{Z}_{2k}$. Based on those simple current modules, a vertex operator
algebra associated with a $\mathbb{Z}_{2k}$-code is constructed. The
classification of irreducible modules for such a vertex operator algebra is
established. Furthermore, all the irreducible modules are realized in a module
for a certain lattice vertex operator algebra.
