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

In this paper, we propose a characterization for non-elementary trapping sets (NETSs) of low-density parity-check (LDPC) codes. The characterization is based on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive. As an application of the proposed characterization/search, we obtain lower and upper bounds on the stopping distance $s_{min}$ of LDPC codes. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $s_{min}$. Finding $s_{min}$, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm

Lower Bounds on the Size of Smallest Elementary and Non-Elementary Trapping Sets in Variable-Regular LDPC Codes

Trapping sets are known to be the main cause for the error floor of low-density parity-check (LDPC) codes. They are often classified by their size a and the number of unsatisfied check nodes b in their subgraph. Trapping sets can be partitioned into two categories of elementary and non-elementary, where the first category are those whose subgraph only contains degree-1 and degree-2 check nodes. Empirical results have shown that often the most harmful trapping sets are elementary. In this letter, we derive a lower bound on the size of the smallest non-elementary trapping sets for a given b in variable-regular LDPC codes. The derived lower bound demonstrates that the size of the smallest possible non-elementary trapping set is, in general, larger than that of an elementary trapping set with the same b value. This provides a theoretical justification as to why non-elementary trapping sets are often not among the most harmful trapping sets

Erratum: On characterization of elementary trapping sets of variable-regular LDPC codes (IEEE Transactions on Information Theory (2015) 61:3)

In the above paper, [1], there are some erroneous entries in Tables I, III, IV, VII, and X, which are corrected here. Moreover, for the proper application of the definition of layered superset (LSS) property to all the results of Tables I-VII in the above-mentioned paper, the LSS definition needs to be extended as described here

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 low-density 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

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

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 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: depth-one tree (dot), path and lollipop, thus, the terminology dpi characterization. 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 ≤ amax and b ≤ bmax, exhaustively. 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

An efficient exhaustive search algorithm for elementary trapping sets of variable-regular LDPC codes

In this paper, we propose an efficient exhaustive search algorithm for elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. Recently, Karimi and Banihashemi proposed a characterization of ETSs, which was based on viewing an ETS 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 ETSs with LSS structure efficiently. Compared to the LSS-based search, which is based on a single LSS expansion technique, the new search algorithm involves two additional expansion techniques. The introduction of the new techniques results in significant improvements in search efficiency compared to the LSS-based search. We prove that using the three expansion techniques, each and every ETS structure can be obtained starting from a simple cycle. We also provide extensive simulation results that show, compared to the LSS-based search, up to three orders of magnitude improvement in search speed and memory requirements can be achieved

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

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 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: depth-one tree (dot), path and lollipop, thus, the terminology dpi characterization. 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 ≤ amax and b ≤ bmax, exhaustively. 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

Tight Lower and Upper Bounds on the Minimum Distance of LDPC Codes

In this paper, we obtain lower and upper bounds on the minimum distance dmin of low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero codewords of an LDPC code into two categories of elementary and non-elementary. The first category contains codewords whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary codewords of an LDPC code with weight less than a certain value amax, exhaustively. We also derive a lower bound Lne on the weight of non-elementary codewords. By performing the search with amax = Lne, we either obtain an elementary codeword with the smallest weight dmin, or establish the lower bound of Lne on dmin. For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on dmin. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, dmin. Finding dmin, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm

Minimal characterization and provably efficient exhaustive search algorithm for elementary trapping sets of variable-regular LDPC codes

In this paper, we propose a new characterization and an efficient exhaustive search algorithm for elementary trapping sets (ETS) of variable-regular low-density parity-check (LDPC) codes. Recently, Karimi and Banihashemi proposed a characterization of ETSs, which was based on viewing an ETS as a layered superset (LSS) of a short cycle in the code's Tanner graph. 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 two problems that LSS-based characterization/search suffers from: (1) exhaustiveness: not every ETS structure is an LSS of a cycle, (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 ETS 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 to exhaustively cover all the ETS structures within the (a, b) classes with a ≤ amax, b ≤ bmax, for any value of amax and bmax, the maximum length of the required cycles is minimal. The proposed characterization corresponds to a provably efficient search algorithm, significantly more efficient than the LSS-based search