Linear network coding

TLDR

A communication network with source nodes multicasting information via multihop, where each node can forward any received data. The study aims to determine the information rate at which each node can receive complete data. The authors optimize multicast by allowing nodes to encode received data, focusing on linear coding where data blocks are treated as vectors over a base field and transformed linearly before forwarding. They prove that linear coding achieves the optimal multicast rate equal to the max‑flow from source to each receiver.

Abstract

Consider a communication network in which certain source nodes multicast information to other nodes on the network in the multihop fashion where every node can pass on any of its received data to others. We are interested in how fast each node can receive the complete information, or equivalently, what the information rate arriving at each node is. Allowing a node to encode its received data before passing it on, the question involves optimization of the multicast mechanisms at the nodes. Among the simplest coding schemes is linear coding, which regards a block of data as a vector over a certain base field and allows a node to apply a linear transformation to a vector before passing it on. We formulate this multicast problem and prove that linear coding suffices to achieve the optimum, which is the max-flow from the source to each receiving node.