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This paper presents an algorithm for computing a feedback law which guarantees that any initial state of the closed loop of a linear time-invariant discrete system will be driven to the origin in minimum finite time. The feedback law is constructed by solving a sequence of algebraic linear equations. The designer has considerable freedom in choosing the generalized eigenvectors which would allow the shaping of the transient response of the system. The algorithm offers, as a byproduct, a new and easy method for computing the so-called "generalized control form" of any controllable pair. Several examples are worked out to demonstrate the feasibility of the proposed method.