Insufficiency of Linear Coding in Network Information Flow

Abstract

Abstract-It is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. It is also known that this result does not generalize to arbitrary networks. There are several examples in the literature of solvable networks with no scalar linear solution over any finite field. However, each example has a linear solution for some vector dimension greater than one. It has been conjectured that every solvable network has a linear solution over some finite-field alphabet and some vector dimension. We provide a counterexample to this conjecture. We also show that if a network has no linear solution over any finite field, then it has no linear solution over any finite commutative ring with identity. Our counterexample network has no linear solution even in the more general algebraic context of modules, which includes as special cases all finite rings and Abelian groups. Furthermore, we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field (exactly 10% greater), so the network is not even asymptotically linearly solvable. It follows that, even for more general versions of linearity such as convolutional coding, filter-bank coding, or linear time sharing, the network has no linear solution.

Network Coding for Speedup in Switches

“…This has led Médard et al [14] to conjecture that every solvble network has a linear solution over some finite field alphabet and vector dimensions. However, Dougherty et al [5] provide a counterexample non-multicast network which is not solvable with linear coding.…”

“…It is important to note that although [5] proves that linear network coding is not sufficient for all networks, linear network coding nevertheless is still a powerful tool.…”

“…In particular, the example network in [5] is extremely complex and contrived, and for our general purpose, we can consider this kind of networks to be outliers, and thus, linear network coding is most of the time sufficient. Therefore, in this thesis, we shall focus on linear network coding and its benefit in a special set of networks called multicast switches (see Section 2.3) which will hopefully bring insight to that of general networks.…”

Wireless Inter-Session Network Coding - An Approach Using Virtual Multicasts

“…We show the benefits of our technique for several scenarios by means of simulation. The optimal solution for inter-session network coding might require nonlinear encoding functions [6] and even deciding a linear inter-session coding problem can be NP-complete [7]. Nonetheless, suboptimal linear coding schemes have been proposed both for wireline and wireless networks.…”

“…For single source network coding problems necessary conditions are also sufficient [1] and distributed linear coding schemes are available [5]. Multi-source network coding is a considerably more difficult problem: The optimal solution for inter-session network coding might require nonlinear encoding functions [6] and even deciding a linear inter-session coding problem can be NP-complete [7]. Nonetheless, suboptimal linear coding schemes have been proposed both for wireline and wireless networks.…”