Optimal Streaming Erasure Codes over the Three-Node Relay Network

This paper investigates low-latency streaming codes for a three-node relay network. The source transmits a sequence of messages (streaming messages) to the destination through the relay between them, where the first-hop channel from the source to the relay and the second-hop channel from the relay to the destination are subject to packet erasures. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay of $T$ time slots. In any sliding window of $T+1$ time slots, we assume no more than $N_1$ and $N_2$ erasures are introduced by the first-hop channel and second-hop channel respectively. Under this channel loss assumption, we fully characterize the maximum achievable rate in terms of $T$, $N_1$ and $N_2$. The achievability is proved by using a symbol-wise decode-forward strategy where the source symbols within the same message are decoded by the relay with different delays. The converse is proved by analyzing the maximum achievable rate for each channel when the erasures in the other channel are consecutive (bursty). In addition, we show that traditional message-wise decode-forward strategies, which require the source symbols within the same message to be decoded by the relay with the same delay, are sub-optimal in general.