On the capacity of non-coherent network coding

Abstract

Abstract-We consider the problem of multicasting information from a source to a set of receivers over a network where intermediate network nodes perform randomized linear network coding operations on the source packets. We propose a channel model for the noncoherent network coding introduced by Koetter and Kschischang in [6], that captures the essence of such a network operation, and calculate the capacity as a function of network parameters. We prove that use of subspace coding is optimal, and show that, in some cases, the capacity-achieving distribution uses subspaces of several dimensions, where the employed dimensions depend on the packet length. This model and the results also allow us to give guidelines on when subspace coding is beneficial for the proposed model and by how much, in comparison to a coding vector approach, from a capacity viewpoint. We extend our results to the case of multiple source multicast that creates a virtual multiple access channel.

Network Coding: Beyond Throughput Benefits

“…To study this, we need to calculate the capacity of noncoherent network coding [21], [22], [24], [25].…”

“…We will use the approach in [21], [22]. We assume communication in time-slots; in each time-slot, the source inserts in the network m packets of length T over some finite field F q , and each receiver collects n packets that consist of random combinations of the source packets.…”

“…To achieve the capacity, a coding scheme may employ the channel (4) multiple times and in this case each codeword is a sequence of input matrices from X . The following theorem characterizes the capacity of noncoherent network coding assuming our channel model [22]. We first state this theorem formally, and then interpret it.…”

Compressed network coding vectors

“…Hence, each source uses a subspace codebook, i.e., maps each message to a set of vectors that span a different subspace. This approach is optimal in terms of achievable information rates when the length of the packets is small but as the length increases it results in the same information rate as the coding vectors approach [3]. Moreover, as the following example illustrates, code design is not trivial when multiple sources insert data in the network.…”

“…From an information theoretic point of view, the second approach, subspace coding, results in higher information rates for short packet length, but as the packet length increases, achieves the same information rate as having the coding vectors overhead [3]. Moreover, it is very challenging to design subspace codes for multisource network coding, where the information sources that insert data in the network are not co-located, as is the case for several applications.…”