A Class of Non-Linearly Solvable Networks

Abstract

For each integer m ≥ 2, a network is constructed which is solvable over an alphabet of size m but is not solvable over any smaller alphabets. If m is composite, then the network has no vector linear solution over any R-module alphabet and is not asymptotically linear solvable over any finite-field alphabet. The network's capacity is shown to equal one, and when m is composite, its linear capacity is shown to be bounded away from one for all finite-field alphabets.

A Local Perspective on the Edge Removal Problem

“…Specifically, we ask if for any given edge e * the non-linear ( , R, n)-feasible network code presented in these works can be modified to give an ( , R, n)-feasible scheme in which the encoding on edge e * is CWL, implying that the CWL statement holds for the instance and rate vector under study. In the case studies from [14]- [16] we are able to modify the non-linear coding scheme appropriately, thus supporting the CWL statement. However we are not able to prove (or disprove) the same for the instance given in [17].…”

“…As any instance for which linear encoding functions are optimal satisfies the conditions of the question above, we study instances for which linear encoding functions are suboptimal. While we do not resolve the question in this work, we prove that such code modifications are possible for the network coding instances and solutions presented in [14]- [16] for which linear coding is known to be sub-optimal, implying that edge removal holds for these solutions. We note that our question is not resolved on the network instance given in [17] for which, as above, linear network coding is suboptimal.…”

“…With our definitions, for any g e ∈ G e , we have φ −1 ge (g e ) = {A α (g e,α ) × Aᾱ(g e,ᾱ ) : g e,α • g e,ᾱ = g e }. (16) As i e ∈ Y α for any α ⊆ S and as i e • i e = i e , by (16)…”

“…C. The network instance of [16] In [16], a class of networks which is solvable but not linearly solvable is introduced. Using the notation of [16], a In B(m) the only encoding edges are the edges carrying message e 0 , e 1 , .…”

“…, e m , e are directly connected to sources such that the local and global encoding function on these edges are identical (as the incoming edge messages are the source messages). Now we show, for any edge in the network N 4 defined in [16] that we can modify the code of [16] and make the global encoding function φ ge CWL. 1) Subnetwork: N 1 (m): In the coding scheme of [16] all edges in N 1 (m) have a linear global encoding function.…”