Motivated by the objective to further reduce the storage and computation time required by finite-horizon discrete-time optimal reduced-order LQG compensators, this paper introduces the modified reachability canonical form of a minimal finite-horizon discrete-time compensator and an algorithm to compute it. Next recursive algorithms for efficient storage and recovery of the compensator matrices in the modified reachability canonical form are presented. Finally the modified reachability grammian, which is associated to the modified reachability canonical form, is used to show and explain why in the finite-horizon case, if the initial compensator state is non-zero, minimality in general does not imply reachability of the compensator.