Trapping sets significantly influence the performance of low-density parity-check codes. An $ (a, b) $ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $ a $ and $ b $ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $ (3, n) $-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected $ (3, n) $-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of $ (a, b) $ ETSs with $ a\leq5 $ and $ b\leq2 $. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.

中文翻译：

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基于有限域乘子子集的无小尺寸基本陷印集的QC-LDPC构造

陷印集会严重影响低密度奇偶校验码的性能。$（a，b）$基本陷印集（ETS）会导致较高的解码失败率，并对代码的错误底限产生很大影响，其中$ a $和$ b $表示未满足的校验和的大小和数量。 ETS中的节点。长度为6的$（3，n）$常规LDPC码中ETS的最小大小为4。在本文中，我们提供了充分的条件来构建完全连接的$（3，n）$常规基于代数的QC-围长为6的LDPC代码，其Tanner图不含$（a，b）$ ETS，以及$ a \ leq5 $和$ b \ leq2 $。我们将这些充分的条件应用于新的基于代数的QC-LDPC码的指数矩阵，其周长至少为6。结果，我们获得了满足足够条件的指数矩阵子矩阵的最大大小，并得出了不含那些小尺寸ETS的Tanner图。文献中一些基于周长为6的基于代数的QC-LDPC代码构造是我们构造的特殊情况。我们的实验结果表明，删除小尺寸的ETS有助于在错误基底区域获得更好的性能曲线。