# American Institute of Mathematical Sciences

August  2020, 14(3): 397-411. doi: 10.3934/amc.2020062

## QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field

 1 Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, 1591634312, Iran 2 School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada

Received  December 2018 Revised  August 2019 Published  August 2020 Early access  January 2020

Fund Project: The authors were partially funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada

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.

Citation: Farzane Amirzade, Mohammad-Reza Sadeghi, Daniel Panario. QC-LDPC construction free of small size elementary trapping sets based on multiplicative subgroups of a finite field. Advances in Mathematics of Communications, 2020, 14 (3) : 397-411. doi: 10.3934/amc.2020062
##### References:

show all references

##### References:
A (5, 3) EAS with $\gamma = 4$ and its corresponding variable node graph
The variable node graphs of $(4, 0)$, $(4, 2)$ and $(5, 1)$ ETSs with girth 6
The comparison of the performance curves of two $(3, 4)$-regular QC-LDPC codes with the same length. The exponent matrices of both codes, $C1$ and $C2$, are submatrices of B in (10)
Row indices $(i, j, k);\ i, j, k\in\{0, 1, 2, 3, 4\}$ and column indices $(c_1, c_2, c_3, c_4);\ c_i\in\{0, 1, \dots, 16\}$ of ${\mathbf B}$ in (10) to construct non-isomorphic $(3, 4)$-regular QC-LDPC codes with girth 6 and free of $(a, b)$ ETSs with $a\leq5$ and $b\leq2$
 $row\ indices$ $column\ indices$ $(1, 2, 3)$ $(1, 2, 7, 10), \ (1, 3, 4, 13), \ (1, 3, 4, 14), \ (1, 3, 13, 14)$ $(1, 2, 3)$ $(1, 4, 5, 16), \ (1, 5, 8, 16), \ (1, 5, 10, 16), \ (1, 5, 12, 15)$
 $row\ indices$ $column\ indices$ $(1, 2, 3)$ $(1, 2, 7, 10), \ (1, 3, 4, 13), \ (1, 3, 4, 14), \ (1, 3, 13, 14)$ $(1, 2, 3)$ $(1, 4, 5, 16), \ (1, 5, 8, 16), \ (1, 5, 10, 16), \ (1, 5, 12, 15)$

2021 Impact Factor: 1.015