XOR-based coding for the 3-user broadcast erasure channel with feedback

We study the case of the three-user broadcast erasure channel, with multiple unicast traffic, where feedback from the users is fed back to the transmitter in the form of ACK/NACK messages. The capacity region of this system has been recently derived and two capacity achieving algorithms, employing intersession linear network coding, have been proposed in [1], [2]. Since these algorithms suffer from large computational complexity and decoding delay, our aim in this paper is to design coding algorithms with reduced computational complexity and low decoding delay that achieve comparable throughput to the former algorithms. We exclusively consider algorithms that require no knowledge of channel statistics, perform only XOR operations between the packets and allow for instantaneous decoding by any receiver that successfully receives a packet. We present two algorithms with the above properties: the first one, named XOR1, operates on a specially constructed network of virtual queues and is seen via simulations to perform adequately with respect to a tight capacity outer bound. The second algorithm, named XOR2, is an enhanced version of XOR1 that operates on the same virtual network and achieves higher throughput through more intelligent packet combining. Furthermore, we show that XOR2 achieves capacity under a general condition, which is satisfied in the following settings: 1) spatially iid erasure channels with arbitrary values of erasure probability, and 2) spatially independent erasure channels where the maximum erasure probability does not exceed 8/9.