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We consider multi--hop networks comprising Binary Symmetric Channels($\mathsf{BSC}$s). The network carries unicast flows for multiple users. Theutility of the network is the sum of the utilities of the flows, where theutility of each flow is a concave function of its throughput. Given that thenetwork capacity is shared by the flows, there is a contention for networkresources like coding rate (at the physical layer), scheduling time (at the MAClayer), etc., among the flows. We propose a proportional fair transmissionscheme that maximises the sum utility of flow throughputs subject to the rateand the scheduling constraints. This is achieved by {\em jointly optimising thepacket coding rates of all the flows through the network}.

Utility Optimal Coding for Packet Transmission over Wireless Networks - Part I: Networks of Binary Symmetric Channels