任意图上线性码实现的约束复杂度
Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs
摘要：
线性码C的图形实现包括将C的坐标分配给图的顶点，以及分别与图的边和顶点相关联的线性状态空间和线性“局部约束”码的规范。图实现的$\k$ -复杂度定义为任意局部约束码的最大维数。$\k$ -复杂度是由图形实现指定的和积解码算法的计算复杂性的合理度量。本文主要研究以下问题：给定一个线性码C和一个图G，C在G上实现的$\k$ -复杂度能有多小？作为解决这个问题的有用工具，我们引入了顶点割界，以及图的“VC树宽”的概念，这与著名的图论树宽概念密切相关。使用这些工具，我们得到严格的下限上的$\k$ -复杂性的任何实现的C在G。我们的边界使我们能够得出结论，良好的纠错码可以有低复杂度的实现，只有在图与大VC树宽度。在此过程中，我们还证明了一个有趣的结果，即长度为n的码C的最佳常规格形实现的$\k$ -复杂度与C的最佳无环实现的$\k$ -复杂度之比最多随码长n而几何地增长。事实上，这种对数增长率是可以实现的。
英文摘要：
A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear ``local constraint'' codes to be associated with the edges and vertices, respectively, of the graph. The $\k$-complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. $\k$-complexity is a reasonable measure of the computational complexity of a sum-product decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the $\k$-complexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the Vertex-Cut Bound, and the notion of ``vc-treewidth'' for a graph, which is closely related to the well-known graph-theoretic notion of treewidth. Using these tools, we derive tight lower bounds on the $\k$-complexity of any realization of C on G. Our bounds enable us to conclude that good error-correcting codes can have low-complexity realizations only on graphs with large vc-treewidth. Along the way, we also prove the interesting result that the ratio of the $\k$-complexity of the best conventional trellis realization of a length-n code C to the $\k$-complexity of the best cycle-free realization of C grows at most logarithmically with codelength n. Such a logarithmic growth rate is, in fact, achievable.
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