  For a natural number $c$, a $c$-arrangement is an arrangement of dimension
$c$ subspaces satisfying the following condition: the sum of any subset of the
subspaces has dimension a multiple of $c$. Matroids arising as normalized rank
functions of $c$-arrangements are also known as multilinear matroids. We prove
that it is algorithmically undecidable whether there exists a $c$ such that a
given matroid has a $c$-arrangement representation, or equivalently whether the
matroid is multilinear. It follows that certain network coding problems are
also undecidable. In the proof, we introduce a generalized Dowling geometry to
encode an instance of the uniform word problem for finite groups in matroids of
rank three. The $c$-arrangement condition gives rise to some difficulties and
their resolution is the main part of the paper.
