This paper deals with stability and state feedback control of discrete-time Multi-Mode Multi-Dimensional (M3D) linear systems. The M3D switch dynamics are modeled through a state mapping describing the mode transitions. This M3D model framework then allows to consider polyquadratic Lyapunov functions to obtain Linear Matrix Inequalities conditions for stability proof and for the synthesis of state-feedback controllers under H∞ performance. A numerical example illustrates the improvement of the controller synthesis conditions here introduced for discrete-time M3D systems over independent point-wise mode solutions.