Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids

We study matroidal networks introduced by Dougherty et al., who showed that if a network is scalar-linearly solvable over some finite field, then the network is a matroidal network associated with a representable matroid over a finite field. In this paper, we prove the converse. It follows that a network is scalar-linearly solvable if and only if the network is a matroidal network associated with a representable matroid over a finite field and that determining scalar-linear solvability of a network is equivalent to finding a representable matroid over a finite field and a valid network-matroid mapping. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields. We note that this result, combined with the construction method due to Dougherty et al., can generate potentially new scalarlinearly solvable networks.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Anthony Kim, Muriel Médard
• Keywords: CORR 2010, Dougherty Et, Education, Finite Fields, Matroidal Network