On the Message Dimensions of Vector Linearly Solvable Networks

TL;DR

This paper investigates the message dimension requirements for vector linear solutions in networks, demonstrating that certain networks require a message dimension at least m to admit a solution, with no solutions at lower dimensions.

Contribution

It generalizes previous results by showing the existence of networks that need a message dimension of at least m for vector linear solutions, extending understanding of message dimension constraints.

Findings

Existence of networks requiring message dimension m for solutions

No vector linear solutions over finite fields for message dimensions less than m

Generalization of previous scalar and 2-dimensional results

Abstract

It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is $2$ [3]. It is not known whether this result can be generalized for an arbitrary message dimension. In this paper, we show that there exists a network which admits an $m$ dimensional vector linear solution, where $m$ is a positive integer greater than or equal to $2$, but does not have a vector linear solution over any finite field when the message dimension is less than $m$.