Analytical lower bounds for the size of elementary trapping sets of variable-regular LDPC codes with any girth and irregular ones with girth 8

In this paper we give lower bounds on the size of (a, b) elementary trapping sets (ETSs) belonging to variable-regular LDPC codes with any girth, g, and irregular ones with girth 8, where a is the size, b is the number of degree-one check nodes and satisfy the inequality b a < 1. Our proposed lower bounds are analytical, rather than exhaustive search-based, and based on graph theories. The numerical results in the literarture for g = 6, 8 for variable-regular LDPC codes match our results. Some of our investigations are independent of the girth and rely on the variables a, b and γ, the column weight value, only. We prove that for an ETS belonging to a variable-regular LDPC code with girth 8 we have a ≥ 2γ − 1 and b ≥ γ. We demonstrate that these lower bounds are tight, making use of them we provide a method to achieve the minimum size of ETSs belonging to irregular LDPC codes with girth 8 specially those whose column weight values are a subset of {2, 3, 4, 5, 6}. Moreover, we show for variable-regular LDPC codes with girth 10, a ≥ (γ − 1) + 1. And for γ = 3, 4 we obtain a ≥ 7 and a ≥ 12, respectively. Finally, for variable-regular LDPC codes with girths g = 2(2k + 1) and g = 2(2k + 2) we obtain a ≥ (γ − 2) + 1 and a ≥ 2(γ − 2) + 1, respectively.