Characterization of Problematic Graphical Structures of LDPC Codes and the Corresponding Efficient Search Algorithms

Abstract

In this thesis, we study all the problematic graphical structures which play important roles in the error floor performance and error correction capability of low-density parity-check (LDPC) codes. These graphical structures are: leafless elementary trapping sets (LETSs), elementary trapping sets (ETSs), non-elementary trapping sets (NETSs), stopping sets and codewords. Recently, Karimi and Banihashemi proposed a characterization of LETSs, which was based on viewing a LETS as a layered superset (LSS) of a short cycle in the code's Tanner graph. However this characterization was not exhaustive. In this work, first, we complement LSS characterization by demonstrating how the remaining structures of LETSs can be characterized. Then, we propose a new characterization for LETSs of variable-regular LDPC codes. Compared to the LSS-based characterization, which is based on a single LSS expansion technique, the new characterization involves two additional expansion techniques. The introduction of the new techniques mitigates the search efficiency problem that LSS-based characterization/search suffers from. 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 the length of the short cycles required to be enumerated 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.Moreover, we generalize the proposed approach of variable-regular to irregular LDPC codes. We explain how the characterization of LETS structures in variableregular graphs can be used to characterize the LETS structures of irregular graphs (in a given range of a and b values, exhaustively). This characterization corresponds to an exhaustive search algorithm. However, this approach is not applicable to irregular LDPC codes, if we are interested to find LETSs in a relatively wide range of iii First, I would like to thank my supervisor, Professor Amir H. Banihashemi for leading and encouraging me during my research. His guidance helped me in all the time of research and writing of this thesis.Thanks must also go to my committee members for their helpful suggestions and discussions.