Rate-Optimal Streaming Codes Over the Three-Node Decode-And-Forward Relay Network

In this paper, we study the three-node Decode-and-Forward (D&F) relay network subject to random and burst packet erasures. The source wishes to transmit an infinite stream of packets to the destination via the relay. The three-node D&F relay network is constrained by a decoding delay of T packets, i.e., the packet transmitted by the source at time i must be decoded by the destination by time i + T . For the individual channels from source to relay and relay to destination, we assume a delay-constrained sliding-window (DCSW) based packet-erasure model that can be viewed as a tractable approximation to the commonly-accepted Gilbert-Elliot channel model. Under the model, any time-window of width w contains either up to a random erasures or else erasure burst of length at most b (≥ a). Thus the source-relay and relay-destination channels are modelled as (a
1
, b
1
, w
1
, T
1
) and (a
2
, b
2
, w
2
, T
2
) DCSW channels. We first derive an upper bound on the capacity of the three-node D&F relay network. We then show that the upper bound is tight for the parameter regime: max{b
1
, b
2
} | (T − b
1
− b
2
− max {a
1
, a
2
} + 1) by constructing streaming codes achieving the bound. The code construction requires field size linear in T , and has decoding complexity equivalent to that of decoding an MDS code.