  We present a study of $B$-parameters for generic $\Delta S=2$ four-quark
operators in domain wall QCD. Our calculation covers all the $B$-parameters
required to study the neutral kaon mixing in the standard model (SM) and beyond
it. We evaluate one-loop renormalization factors of the operators employing the
plaquette and Iwasaki gauge actions. Numerical simulations are carried out in
quenched QCD with both gauge actions on $16^3\times 32\times 16$ and
$24^3\times 32\times 16$ at the lattice spacing $1/a\approx 2$GeV. We
investigate the relative magnitudes of the non-SM $B$-parameters to the SM one,
which are compared with the previous results obtained with the overlap and the
clover quark actions.
