In this paper, a new and simple method for the construction of Girth-6 (J,L) Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes is proposed. The method is further extended to the search of Girth-8 QC-LDPC codes with base matrices of order 3 × L and 4 × L. The construction is based on three different forms of exponent matrices and the parameters α, p, and q which satisfy the necessary algebraic conditions for a QC-LDPC code having girth 6 and 8. The proposed (J,L) QC-LDPC codes with girth at least six have optimal circulant permutation matrix (CPM) size for the cases where q = α + 1. Moreover, most of the girth-8 QC-LDPC codes searched by the proposed method have smaller CPM size than the existing codes of the same girth. In several cases, the method gives more than one exponent matrices for a code, as most of the existing methods cannot do so. Besides this, the proposed method not only search the QC-LDPC codes with smaller CPM size but also takes much less time than the existing search based methods to search code.

中文翻译：

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搜索围长为6和8的最小QC-LDPC码

本文提出了一种新的构造Girth-6 （J，L）准循环低密度奇偶校验（QC-LDPC）码的简单方法。该方法进一步扩展到搜索基序为3× L和4×  L的Girth-8 QC-LDPC码 。该构造基于三种不同形式的指数矩阵以及参数 α，p和q，它们满足具有6和8围长的QC-LDPC码的必要代数条件。提出的（J，L） QC-LDPC码具有对于q  =  α的情况，至少六个周长具有最佳循环置换矩阵（CPM）大小 +1。此外，通过该方法搜索的大多数第8围QC-LDPC码的CPM大小均小于相同围的现有码。在某些情况下，该方法为代码提供了多个指数矩阵，因为大多数现有方法无法做到这一点。除此之外，所提出的方法不仅搜索具有较小CPM大小的QC-LDPC码，而且比现有的基于搜索的方法来搜索代码所花费的时间要少得多。