#### Title

Optimal Linear Network Coding When 3 Nodes Communicate Over Broadcast Erasure Channels with ACK

#### Abstract

This work considers the following scenario: Three nodes {1, 2, 3} would like to communicate with each other by sending packets through unreliable wireless medium.We consider the most general unicast traffic demands. Namely, there are six co-existing unicast flows with rates (R_{1-->2},R_{1-->3},R_{2-->1},R_{2-->3}, R_{3-->1},R_{3-->2}). When a node broadcasts a packet, a random subset of the other two nodes will receive the packet. After each transmission, causal ACKnowledgment is sent so that all nodes know whether the other nodes have received the packet or not. Such a setting has many unique features. For example, each node, say node 1, can assume many different roles: Being the transmitter of the information R_{1-->2} and R_{1-->3}; being the receiver of the information R_{2-->1} and R_{3-->1}; and being the relay for the information R_{2-->3} and R_{3-->2}. This fully captures the fundamental behaviors of 3-node network communications. Allowing network coding (NC) to capitalize the diversity gain (i.e., overhearing packets transmitted by other nodes), this work characterizes the 6-dimensional linear network coding (LNC) capacity of the above erasure network. The results show that for any channel parameters, the LNC capacity can be achieved by a simple strategy that involves only a few LNC choices.

#### Date of this Version

6-24-2014