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Time-slotted ALOHA packetized multiple-access has been extensively used in satellite communications, and has recently been attracting considerable attention in wireless Ad- hoc networks. In this paper, we consider a time-slotted ALOHA system which combines multiple-access, broadcasting channels and rate splitting. This system allows some transmission bits to be reliably received even when collisions occur and more bits to be reliably received in the absence of collisions. In contrast to previous work, our work focuses on the case in which the transmission channels obey i.i.d (independent and identically distributed) Rayleigh fading. Two fundamental problems are considered. The first one is the calculation of the system capacity, and the second one is how to select an appropriate channel coding scheme with which the system achieves its capacity. We shall review the single time slotted capacity, and present the total capacity expression with the knowledge of channel side information. We will then derive a threshold for the transmission probability in a single time slot provided that all the users have the same transmission probability. It will be shown that when the transmission probability is greater than the derived threshold, low-resolution codes can help the system achieve its capacity. We shall also present an explicit estimation method for computing the threshold, and prove that it can be extended to the more general case when the transmission probabilities are approximately equal.