This paper investigates the problem of communication scheduling probability and controller co-design for Markov jump systems (MJSs) based on fading networks. Specifically, only one channel of the fading signal can be transmitted at each transmission moment. The signal from the sensors is transmitted to the controller via the fading network. In order to plan the channel selection, a stochastic communication protocol (SCP) is introduced. The channel selection for the SCP and the fading coefficients in the fading network are modeled as Markov models. Using the Lyapunov method, sufficient conditions that simultaneously satisfy the exponential mean-square stability of the closed-loop system and the preset $H_{\infty}$ performance are derived. However, the developed stability and controller design conditions include nonlinear constraints. To handle nonlinear constraints and achieve optimal closed-loop system $H_{\infty}$ performance, this paper proposes an iterative algorithm that combines a genetic algorithm with penalty constraints and a linear matrix inequality method. A selection scheme for the initial value of the iterative algorithm is also provided. Finally, the effectiveness of the proposed control method is demonstrated.