This article addresses the problem of distributed controller design for linear discrete-time systems. The problem is posed using the classical framework of state feedback gain optimization over an infinite-horizon quadratic cost, with an additional sparsity constraint on the gain matrix to model the distributed nature of the controller. An equivalent formulation is derived that consists in the optimization of the steady-state solution of a matrix difference equation, and two algorithms for distributed gain computation are proposed based on it. The first method consists in a step-by-step optimization of said difference matrix equation, and allows for fast computation of stabilizing state feedback gains. The second algorithm optimizes the same matrix equation over a finite time window to approximate asymptotic behavior and thus minimize the infinite-horizon quadratic cost. To assess the performance of the proposed solutions, simulation results are presented for the problem of distributed control of a quadruple-tank process, as well as a version of that problem scaled up to 40 interconnected tanks.