  It is known that the Fano network has a vector linear solution if and only if
the characteristic of the finite field is $2$; and the non-Fano network has a
vector linear solution if and only if the characteristic of the finite field is
not $2$. Using these properties of Fano and non-Fano networks it has been shown
that linear network coding is insufficient. In this paper we generalize the
properties of Fano and non-Fano networks. Specifically, by adding more nodes
and edges to the Fano network, we construct a network which has a vector linear
solution for any vector dimension if and only if the characteristic of the
finite field belongs to an arbitrary given set of primes
$\{p_1,p_2,\ldots,p_l\}$. Similarly, by adding more nodes and edges to the
non-Fano network, we construct a network which has a vector linear solution for
any vector dimension if and only if the characteristic of the finite field does
not belong to an arbitrary given set of primes $\{p_1,p_2,\ldots,p_l\}$.
