Characterization of Elementary Trapping Sets in Irregular LDPC Codes and the Corresponding Efficient Exhaustive Search Algorithms

In this paper, we propose a characterization of elementary trapping sets (ETSs) for irregular low-density parity-check (LDPC) codes. These sets are known to be the main culprits in the error floor region of such codes. The characterization of ETSs for irregular codes has been known to be a challenging problem due to the large variety of non-isomorphic ETS structures that can exist within the Tanner graph of these codes. This is a direct consequence of the variety of the degrees of the variable nodes that can participate in such structures. The proposed characterization is based on a hierarchical graphical representation of ETSs, starting from simple cycles of the graph, or from single variable nodes, and involves three simple expansion techniques: degree-one tree (<inline-formula> <tex-math notation="LaTeX">$dot$ </tex-math></inline-formula>), <inline-formula> <tex-math notation="LaTeX">$path$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$lollipop$ </tex-math></inline-formula>, thus, the terminology <italic>dpl characterization</italic>. A similar <inline-formula> <tex-math notation="LaTeX">$dpl$ </tex-math></inline-formula> characterization was proposed in an earlier work by the authors for the leafless ETSs of variable–regular LDPC codes. The present paper generalizes the prior work to codes with a variety of variable node degrees and to ETSs that are not leafless. The proposed <inline-formula> <tex-math notation="LaTeX">$dpl$ </tex-math></inline-formula> characterization corresponds to an efficient search algorithm that, for a given irregular LDPC code, can find all the instances of <inline-formula> <tex-math notation="LaTeX">$(a, b)$ </tex-math></inline-formula> ETSs with size <inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> and with the number of unsatisfied check nodes <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> within any range of interest <inline-formula> <tex-math notation="LaTeX">$a \leq a_{\max }$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$b \leq b_{\max }$ </tex-math></inline-formula>, exhaustively. Although branch-&-bound exhaustive search algorithms for finding ETSs of irregular LDPC codes exist, to the best of our knowledge, the proposed search algorithm is the first of its kind, in that, it is devised based on a characterization of ETSs that makes the search process efficient. For a constant degree distribution and range of search, the worst-case complexity of the proposed <inline-formula> <tex-math notation="LaTeX">$dpl$ </tex-math></inline-formula> algorithm increases linearly with the block length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. The average complexity, excluding the search for the input simple cycles, is constant in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Extensive simulation results are presented to show the versatility of the search algorithm, and to demonstrate that, compared to the literature, significant improvement in search speed can be obtained.