平面图上的码
Codes on Planar Graphs
摘要：
定义在图上的码及其性质是最近研究的热点。在实践方面，容量逼近码的构造是图形化的。在理论方面，图上的编码在编码理论和图论的交叉中提供了几个有趣的问题。本文研究了由平面Tanner图定义的码。我们推导出此类代码的最小距离$d$的上界，作为$R \ge 5/8$的码率$R$的函数。这个结果的有趣结论如下：（1）平面图不支持渐近好码，（2）在具有高最小距离的图上的有限长度、高速率码必然是非平面的。
英文摘要：
Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several intriguing problems in the intersection of coding theory and graph theory. In this paper, we study codes defined by planar Tanner graphs. We derive an upper bound on minimum distance $d$ of such codes as a function of the code rate $R$ for $R \ge 5/8$. The bound is given by $$d\le \lceil \frac{7-8R}{2(2R-1)} \rceil + 3\le 7.$$ Among the interesting conclusions of this result are the following: (1) planar graphs do not support asymptotically good codes, and (2) finite-length, high-rate codes on graphs with high minimum distance will necessarily be non-planar.
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