Here we consider the problem of stabilizing a discrete-time linear time-invariant plant by a linear periodic compensator. We show that we can trade off compensator order for an increased period. In particular, with n the plant order, it is shown that for every l is an element of {1,..., n}, we can do almost arbitrary pole placement using an lth-order, (n + 2 - l)th period, linear periodic controller; for most plants the period can be reduced by one. We also show that first-order linear periodic controllers can be used to solve the strong and simultaneous stabilization problems.