In this paper, the co-design of continuous-variable controllers and discrete-event switching logics, both in state feedback form, is investigated for a class of discrete-time switched linear control systems. It is assumed that none of the subsystems is stabilized through a continuous state feedback alone. However, it is possible to stabilize the whole switched system via carefully designing both the continuous controllers and the switching logics. Sufficient synthesis conditions for this co-design problem are proposed here in the form of bilinear matrix inequalities, which is based on the argument of multiple Lyapunov functions. The closed-loop switched system forms a special class of linear hybrid system, and is shown to be asymptotically stable with a finite l(2) induced gain.