Linear network coding capacity for broadcast erasure channels with feedback, receiver coordination, and arbitrary security requirement

This work considers a commonly encountered wireless transmission scenario. The base station s would like to send two independent packet streams to clients d₁ and d₂, respectively. For each time slot, only one of the three nodes {s, d₁, d₂} can transmit a packet and the packet will be heard by a random subset of the other two nodes. We are interested in the corresponding capacity region (R₁*, R₂*). Such a setting can also be viewed as allowing receiver coordination for the s-to-{d₁, d₂} broadcast erasure channel with a critical feature that any coordination/transmission between d₁ and d₂ also takes away the precious time resources from s. With the exclusive focus on linear network coding (LNC) with causal packet acknowledgement feedback, this work characterizes the exact LNC capacity region with arbitrary security requirement, i.e, the system designer can decide for each d_i, respectively, whether the corresponding (s, d_i)-flow needs to be secure or not. The results show that for any channel parameters and any security requirement, the LNC capacity can always be achieved either by the XOR-in-the-air LNC scheme, or by random LNC, or by time-sharing between the two.