Multicast Networks Solvable over Every Finite Field

Abstract

In this work, it is revealed that an acyclic multicast network that is scalar linearly solvable over Galois Field of two elements, GF (2), is solvable over all higher finite fields. An algorithm which, given a GF (2) solution for an acyclic multicast network, computes the solution over any arbitrary finite field is presented. The concept of multicast matroid is introduced in this paper. Gammoids and their base-orderability along with the regularity of a binary multicast matroid are used to prove the results.

On NACK-Based rDWS Algorithm for Network Coded Broadcast

“…A network is said to be solvable if each receiver in the network can recover their demands (set of packets, originated at the sources) from the received network coded packets [15]. When any link in a network carries a single data packet (raw or coded) and coding coefficients at each intermediate node are considered as vectors, it is termed as scalar network coding [16].…”

“…The connection of field size with linear solvability of a network was first revealed in Reference [6], where it was shown that a multicast network is linearly solvable if the field size is sufficiently large. Since then, there has been extensive research on the effect of field size on scalar linear solvability and on the effect of field size and message dimension on vector linear solvability of various multicast and non-multicast network configurations [15,[18][19][20][21][22][23][24][25]. Some recent work on this solvability issue has shown that not only the field size but also the order of the subgroups in the multiplicative group of the finite field are responsible for linear solvability of a multicast network [26].…”