准循环LDPC码：原模图和Tanner图结构对最小汉明距离上界的影响
信息论
2016-11-17 v2 离散数学
math.IT
摘要
准循环低密度奇偶校验码是基于原模图的LDPC码的重要实例。本文给出了准循环LDPC码最小汉明距离的上界，并研究了这些上界如何依赖于Tanner图及其底层原模图的图结构参数（如变量节点度数、校验节点度数、围长）。此外，针对几类原模图，我们给出了能够达到（或接近）相应最小汉明距离上界的显式准循环LDPC码构造。由于准循环码与卷积码之间存在紧密的代数联系，我们可以对卷积码的自由汉明距离陈述类似的结果。事实上，一些准循环码的结论是通过首先证明相应的卷积码结论，然后利用Tanner的一个结果（即准循环码的最小汉明距离上界受限于通过“展开”该准循环码得到的卷积码的自由汉明距离）来建立的。
引用
@article{arxiv.0901.4129,
title = {Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds},
author = {Roxana Smarandache and Pascal O. Vontobel},
journal= {arXiv preprint arXiv:0901.4129},
year = {2016}
}
备注
To appear in IEEE Transactions on Information Theory. Changes compared to v1: some convolutional code results have been added; some incompleteness issues with some of the proofs have been corrected; a typo in one of the parity-check matrices has been corrected (i.e., an entry of H"(x) in Example 28 of v1 needs to be changed so that d_min=56 as written there, cf. \hat H(x) in Example 29 of v2)