A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth
A Unifying Framework to Construct QC-LDPC Tanner Graphs of Desired Girth
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DOI：
10.1109/tit.2022.3170331
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发表时间：
2021-08
影响因子：
2.5
通讯作者：
R. Smarandache;David G. M. Mitchell
中科院分区：
文献类型：
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作者：
R. Smarandache;David G. M. Mitchell
本文提出了一个统一的框架来构造低密度奇偶校验（LDPC）码与相关的坦纳图所需的围长。为了实现这一目标，我们强调的作用，出现在产品的奇偶校验矩阵与转置的代码与所需的围长图的建设中的某个方阵，并进一步探讨它，以产生一组必要和充分条件的坦纳图有一个给定的围长在6和12之间。对于每个这样的围长，我们提出了构造所需围长代码的算法，并展示了如何使用它们来计算提升因子的最小必要值。对于周长大于12，我们展示了如何使用多步图提升方法来确定性地修改代码，以增加其周长。我们还给出了一个新的视角LDPC原型为基础的奇偶校验矩阵，将它们视为行的奇偶校验矩阵等于某些置换矩阵的总和，并获得所有原型和那些变量节点的度2之间的重要连接。我们还表明，我们开发的所有一个protograph的结果和方法，可以使用和适应分析周长的任何奇偶校验矩阵的坦纳图，并演示如何使用一个著名的不规则，多边protograph指定的美国宇航局空间数据系统咨询委员会（CCSDS）。通过构造围长在6到14之间的坦纳图的LDPC码，验证了我们的理论结果，并给出了多步提升校验矩阵围长在14到22之间的充分条件.
This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated Tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a Tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the Tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with Tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.