拟阵路径宽度与码格复杂度
Matroid Pathwidth and Code Trellis Complexity
摘要：
我们将拟阵路宽的概念与线性码的最小网格状态复杂度（我们称之为网格宽度）以及图的路宽联系起来。通过将图的路宽计算问题归约，证明了确定可表示拟阵的路宽问题是NP-困难的。因此，计算线性码的网格宽度的问题也是NP难的。对于有限域$\F$，我们还考虑了路径宽度至多为$w$的$\F$ -可表示拟阵类，以及相应的网格宽度至多为$w$的$\F$上的线性码族.这些很容易被视为小封闭。由于这些拟阵（和码）的分支宽度最多为$w$，Geelen和Whittle的一个结果表明，这些拟阵（和相应的码）的特征在于有1000个排除子式。我们提供了$w=1$排除的未成年人的完整列表，并给出了$w=2$的部分列表。
英文摘要：
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 $w=2$.
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