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On the Girth of Quasi-Cyclic Protograph LDPC Codes

机译：准循环原型LDPC码的周长

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In this paper, we study the relationships between the girth of the Tanner graph of a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.
机译：在本文中，我们研究了准循环（QC）原型低密度奇偶校验（LDPC）码的Tanner图的周长，提升度与基础图的大小和结构之间的关系。结果，对于给定的基础图，我们得出提升度的下限，作为提升图具有一定周长的必要条件。这也提供了从给定基础图以给定提升度提升的图族的周长的上限。此处导出的上限适用于没有平行边的规则图和不规则基础图，在某些情况下比现有范围更通用，在其他情况下更紧密。在这项工作中提出的结果可用于设计相对较小的度数和相对较大的周长的周期性起重。例如，我们使用完全连接的基础图展示了围长为8的新QC原型LDPC代码构造。这些构造使用完全连接的基础曲线图提供了达到周长8所需的提升程度的上限。

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