  We study a class of nearest-neighbor minimally doubled actions which depend
on 2 continuous parameters. We calculate the contributions of the 3 possible
counterterms in perturbation theory, and we find that for each counterterm
there are curves in the parameter space on which its coefficient vanishes. One
can thus construct renormalized actions that contain only 2 counterterms
instead of the 3 of the standard Karsten-Wilczek or Borici-Creutz actions.
  Our investigations suggest the usefulness of analogous nonperturbative
searches for values of the parameters for which the number of counterterms can
be reduced. They can also be an inspiration to undertake a search for
ultralocal minimally doubled actions with even better counterterm-reducing
properties, including the optimal case in which all counterterms can be
removed. Simulations of the latter actions will be much cheaper than the cases
in which one needs to add counterterms to the bare actions, like the already
conveniently inexpensive standard Karsten-Wilczek fermions.
  Finally, we also introduce minimally doubled fermions with
next-to-nearest-neighbor interactions, which depend on 4 continuous parameters,
as a further possibility in the search for renormalized actions with no
counterterms.
