Matroid Pathwidth and Code Trellis Complexity

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 $w=2$.