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In asynchronous systems, the sender encodes a data word with a code word from an unordered code and transmits the code word on the parallel bus lines. In this paper, a transmission time analysis for the above parallel asynchronous communication scheme is presented. It is proven that the average transmission time for a code word is a strictly increasing function of the weight of the code word and it approaches the worst transmission time possible when the weight goes to infinity. This implies that fast parallel asynchronous systems can be designed using low weight codes. This paper also analyzes the transmission time performances of the proximity detecting codes and gives some efficient low constant weight code designs.