New Characterization and Efficient Exhaustive Search Algorithm for Leafless Elementary Trapping Sets of Variable-Regular LDPC Codes

In this paper, we propose a new characterization for leafless elementary trapping sets (LETSs) of variable-regular lowdensity parity-check codes. Recently, Karimi and Banihashemi proposed a characterization of LETSs, which was based on viewing an LETS as a layered superset (LSS) of a short cycle in the code's Tanner graph. A notable advantage of LSS characterization is that it corresponds to a simple LSS-based search algorithm (expansion technique) that starts from short cycles of the graph and finds the LETSs with LSS structure efficiently. Compared with the LSS-based characterization of Karimi and Banihashemi, which is based on a single LSS expansion technique, the new characterization involves two additional expansion techniques. The introduction of the new techniques mitigates two problems that LSS-based characterization/search suffers from: 1) exhaustiveness: not every LETS structure is an LSS of a cycle and 2) search efficiency: LSS-based search algorithm often requires the enumeration of cycles with length much larger than the girth of the graph, where the multiplicity of such cycles increases rapidly with their length. We prove that using the three expansion techniques, any LETS structure can be obtained starting from a simple cycle, no matter how large the size of the structure a or the number of its unsatisfied check nodes b are, i.e., the characterization is exhaustive. We also demonstrate that for the proposed characterization/search to exhaustively cover all the LETS structures within the (a, b) classes with a amax and b bmax, for any value of amax and bmax, the length of the short cycles required to be enumerated is less than that of the LSS-based characterization/search. We, in fact, show that such a length for the proposed search algorithm is minimal. We also prove that the three expansion techniques, proposed here, are the only expansions needed for characterization of LETS structures starting from simple cycles in the graph, if one requires each and every intermediate sub-structure to be a LETS as well. Extensive simulation results are provided to show that, compared with LSS-based search, significant improvement in search speed and memory requirements can be achieved.