Convolutional Network Codes for Cyclic Networks

Most data networks contain cycles, but so far most attention in the literature of network coding has been addressed to multicast in acyclic networks. The original paper on network coding [1] did consider cyclic networks but there it was suggested to transform the cyclic network into an acyclic network using the idea of unrolling the network into a layered network. This approach has many drawbacks: it achieves the optimal rate only asymptotically, it leads to time-variant schemes, it has high encoding and decoding complexities and it induces large delay. In [2] it was shown that if each edge in the network has delay, then there exists a time-invariant linear network code that achieves the optimal rate. This approach may, again, introduce large delay since each edge has delay and an efficient construction algorithm is not given. In [3] a heuristic code construction is given for a linear time-invariant code, but the construction is not given explicitly and it is not necessarily efficient. In this work we give an explicit polynomial time code construction of an optimal multicast linear network code for cyclic networks. As it turns out, it is not necessary for every edge in the network to have delay, as long as we ensure that in each cycle in the network at least one edge has delay. Since delay elements are anyway inserted into the network it seems more natural to focus on convolutional codes for cyclic networks. Nevertheless, our results in this paper are directly applicable for block codes.