  We relate the notion of matroid pathwidth to the minimum trellis
state-complexity (which we term trellis-width) of a linear code, and to the
pathwidth of a graph. By reducing from the problem of computing the pathwidth
of a graph, we show that the problem of determining the pathwidth of a
representable matroid is NP-hard. Consequently, the problem of computing the
trellis-width of a linear code is also NP-hard. For a finite field $\F$, we
also consider the class of $\F$-representable matroids of pathwidth at most
$w$, and correspondingly, the family of linear codes over $\F$ with
trellis-width at most $w$. These are easily seen to be minor-closed. Since
these matroids (and codes) have branchwidth at most $w$, a result of Geelen and
Whittle shows that such matroids (and the corresponding codes) are
characterized by finitely many excluded minors. We provide the complete list of
excluded minors for $w=1$, and give a partial list for $w=2$.
