Time optimal discrete regulator gains

Given a discrete time system (
A, B
) where
B
has full rank, does there exist a gain matrix
G
such that
(A + BG)^{k}x = 0
for every state
x
that can be driven to the origin in time
k?
If so, how does one compute
G?
Complete answers to these questions are given, and errors in a previous correspondence are corrected.