  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
$m$.
