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In this paper, the linear quadratic (LQ) optimal control of discrete-time linear time-invariant (LTI) systems with random input gains is studied. We define the capacity of each input channel whose sum yields the total capacity of all input channels. Different from the finite-horizon case which can be solved by dynamic programming, the infinite-horizon case may be unsolvable if the capacities of the individual channels are fixed a priori. The main novelty of this work is that we put the problem under the framework of channel/controller co-design which allows the control designer to have the additional freedom to design the channels. We assume that the overall channel capacity is constrained which can be allocated to the individual channels. By channel/controller co-design, it is shown that the infinite-horizon case is solvable if and only if the overall capacity of the input channels is greater than the topological entropy of the open-loop plant. Moreover, the optimal control signal is a linear state feedback.