In this paper, we consider time-varying state feedback control of single-input linear time-invariant discrete-time systems. The approach that we propose is based on a contraction property that was introduced in a previous paper. All time-varying controllers that yield the contraction property are derived. It is seen that the state feedback vector becomes m-periodic, where the integer m is a user parameter related to the obtained contraction property. From the class of control laws that yield the contraction property, we determine the controller which minimizes a linear quadratic criterion. Results that are useful for tuning one of the user parameters are presented. Numerical examples which compare the methods are also included.