关于线性网格的代数结构
On the Algebraic Structure of Linear Trellises
摘要：
网格是代码的重要图形表示。虽然传统的网格是很好地理解，（咬尾）网格的一般理论仍在发展中。迭代解码具体地激发了这种理论。在本文中，我们首先开发了一个新的代数框架的线性网格，使我们能够解决开放的基础问题的系统分析。特别是，我们提出了一个有用的和强大的线性网格同构的特征。我们还得到了Koetter/Vardy分解定理的一个新的证明，并指出了群情形下未被注意的问题。其次，我们利用我们的工作：描述了线性格的所有初等格分解，从而确定了给定码的所有最小线性格;证明了不可合并的一对一线性格是由某些闭路的边标号序列惊人地确定的;证明了最小线性格的自对偶定理;分析准循环线性格架，从而将约化线性格架上的结果扩展到非约化。为了实现这一点，我们还提供了新的见解合并和路径连接属性的线性网格。我们的分类结果是重要的迭代解码，因为我们表明，最小的线性网格可以产生不同的伪码字，即使他们有相同的图形结构。
英文摘要：
Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the Factorization Theorem of Koetter/Vardy and point out unnoticed problems for the group case. Next, we apply our work to: describe all the elementary trellis factorizations of linear trellises and consequently to determine all the minimal linear trellises for a given code; prove that nonmergeable one-to-one linear trellises are strikingly determined by the edge-label sequences of certain closed paths; prove self-duality theorems for minimal linear trellises; analyze quasi-cyclic linear trellises and consequently extend results on reduced linear trellises to nonreduced ones. To achieve this, we also provide new insight into mergeability and path connectivity properties of linear trellises. Our classification results are important for iterative decoding as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure.
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