Characterization and efficient exhaustive search algorithm for elementary trapping sets of irregular LDPC codes

Abstract

In this paper, we propose a characterization of elementary trapping sets (ETSs) for irregular lowdensity 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 (dot), path and lollipop, thus, the terminology dpl characterization. A similar dpl characterization was proposed in an earlier work by the authors for the leafless ETSs (LETSs) 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 dpl characterization corresponds to an efficient search algorithm that, for a given irregular LDPC code, can find all the instances of (a, b) ETSs with size a and with the number of unsatisfied check nodes b within any range of interest a ≤ a max and b ≤ b max , exhaustively. Although, (brute force) exhaustive search algorithms for 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. 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.

Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes With Applications to Stopping Sets

“…The characterization of ETSs (LETSs and ETSLs) for variable-regular graphs, provided in [7] and [8], is based on normal graph representation of structures. This approach, however, is not applicable to NETSs.…”

“…The new algorithm also searches for both ESSs and NESSs, and to search for both categories, it requires to search for LETSs. The LETS search can be performed through the exhaustive dpl searches of [7] and [8], for regular and irregular graphs, respectively. The problem, however, is that the complexity of such a search increases rather rapidly, if the range of search, indicated by the value of a max , is increased much beyond the value of L SS 3 .…”

“…Normal graphs have been used to develop the most efficient search algorithms for ETSs [15], [7], [8].…”

“…Otherwise, the stopping set is referred to as non-elementary (NESS). Considering that an ESS is a LETS with no unsatisfied check node, we use the highly efficient algorithms of [7], [8], to search for ESSs. NESSs, on the other hand, are a subset of NETSs.…”

“…Basic definitions and notations are provided in Section II. We also briefly explain the search algorithms of [7] and [8] for finding LETSs/ETSs in this section. Section II ends with revisiting the lower bounds derived in [9] on the smallest size of ETSs and NETSs.…”