线性码的最小树实现
On Minimal Tree Realizations of Linear Codes
摘要：
代码坐标的树分解是从坐标集到树的顶点集的映射。树分解可以扩展到树实现，即，通过在树的每个边缘指定状态空间，以及在树的每个顶点指定局部约束代码，在底层树上实现代码的无环实现。树实现的约束复杂度是它的任何局部约束码的最大维数。最大似然译码复杂度的量度是其树宽，这是其任何树实现的最小约束复杂度。众所周知，在扩展给定树分解的代码的所有树实现中，存在唯一的最小实现，其最小化底层树的每个顶点处的状态空间维度。本文给出了这类极小实现的两个新的构造。作为副产品的第一个建设，一个概括的状态合并过程的网格实现，我们得到的事实是，最小的树实现也最小化的局部约束码的尺寸在每个顶点的底层树。第二种构造依赖于我们开发的某些代码分解技术。我们进一步观察到，代码的树宽与图复杂度的度量有关，也称为树宽。我们利用这种连接来解决Forney的猜想之间的差距的最小网格约束的复杂性和树宽的代码。我们提出了一个家庭的代码，这个差距可以任意大。
英文摘要：
A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations. It is known that among all tree realizations of a code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.
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