LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes.
翻译:从变异矩阵中构建的LDPC代码最近引起了许多研究人员的兴趣。 处理这种代码的一个关键点是试图避免相关的Tanner图中短时间的周期,即获得一个可能大的长长的周期。 在本文中,我们提供了一个框架来获得这种代码的构建。我们把存在一定长的周期的标准与一些数字理论概念相联系,特别是与所谓的Sidon数据集相联系。我们通过这种方式获得某些长长的LDPC代码的例子。 最后,我们扩展我们的构建范围,以获得不规则的LDPC代码。