  We show that computing the Tutte polynomial of a linear matroid of dimension
$k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires
$k^{\Omega(k)}$ time unless the \#ETH---a counting extension of the Exponential
Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.}
[ACM TALG 2014]---is false. This holds also for linear matroids that admit a
representation where every point is associated to a vector with at most two
nonzero coordinates. We also show that the same is true for computing the Tutte
polynomial of a binary matroid of dimension $k$ on $k^{O(1)}$ points with at
most three nonzero coordinates in each point's vector. This is in sharp
contrast to computing the Tutte polynomial of a $k$-vertex graph (that is, the
Tutte polynomial of a {\em graphic} matroid of dimension $k$---which is
representable in dimension $k$ over the binary field so that every vector has
two nonzero coordinates), which is known to be computable in $2^k k^{O(1)}$
time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via
(i) a connection due to Crapo and Rota [1970] between the number of tuples of
codewords of full support and the Tutte polynomial of the matroid associated
with the code; (ii) an earlier-established \#ETH-hardness of counting the
solutions to a bipartite $(d,2)$-CSP on $n$ vertices in $d^{o(n)}$ time; and
(iii) new embeddings of such CSP instances as questions about codewords of full
support in a linear code. We complement these lower bounds with two algorithm
designs. The first design computes the Tutte polynomial of a linear matroid of
dimension~$k$ on $k^{O(1)}$ points in $k^{O(k)}$ operations. The second design
generalizes the Bj\"orklund~{\em et al.} algorithm and runs in
$q^{k+1}k^{O(1)}$ time for linear matroids of dimension $k$ defined over the
$q$-element field by $k^{O(1)}$ points with at most two nonzero coordinates
each.
