  We consider the vector linear solvability of networks over a field
$\mathbb{F}_q.$ It is well known that a scalar linear solution over
$\mathbb{F}_q$ exists for a network if and only if the network is
\textit{matroidal} with respect to a \textit{matroid} representable over
$\mathbb{F}_q.$ A \textit{discrete polymatroid} is the multi-set analogue of a
matroid. In this paper, a \textit{discrete polymatroidal} network is defined
and it is shown that a vector linear solution over a field $\mathbb{F}_q$
exists for a network if and only if the network is discrete polymatroidal with
respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An
algorithm to construct networks starting from a discrete polymatroid is
provided. Every representation over $\mathbb{F}_q$ for the discrete
polymatroid, results in a vector linear solution over $\mathbb{F}_q$ for the
constructed network. Examples which illustrate the construction algorithm are
provided, in which the resulting networks admit vector linear solution but no
scalar linear solution over $\mathbb{F}_q.$
