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 ≤ 5 and b ≤ 2. 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.

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Keywords Algebraic-based QC-LDPC codes, Edge coloring, Elementary trapping set, Girth, Tanner graph
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Journal Advances in Mathematics of Communications
Amirzade, F. (Farzane), Sadeghi, M.-R. (Mohammad-Reza), & Panario, D. (2020). QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field. Advances in Mathematics of Communications, 14(3), 397–411. doi:10.3934/amc.2020062