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 dpl characterization . 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.
The chiral RhI-diene-catalyzed asymmetric three-component reaction of aryldiazoacetates, aromatic amines, and beta-nitroacrylates was achieved to obtain gamma-nitro-alpha-amino-succinates in good ...yields and with high diastereo- and enantioselectivity. This reaction is proposed to proceed through the enantioselective trapping of RhI-associated ammonium ylides by nitroacrylates. This new transformation represents the first example of RhI-carbene-induced ylide transformation.
Asymmetric functionalization of aromatic CH bonds of N,N-disubstituted anilines with diazo compounds and imines is reported for the efficient construction of alpha,alpha-diaryl benzylic quaternary ...stereocenters in good yields with high diastereoselectivities and excellent enantioselectivities. This RhII/chiral phosphoric acid cocatalyzed transformation is proposed to proceed through a metal-carbene-induced zwitterionic intermediate which undergoes electrophilic trapping. To the best of our knowledge, this is the first asymmetric example of metal carbene-induced intermolecular functionalization of aryl CH bonds.
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.
Although the tunnel mass is indispensable to predict the gate leakage current of electron devices, it has been regarded as an adjustable parameter to fit the calculated leakage current with the ...measured ones. This appears useful because it enables calculation of the tunnel current while ignoring some details in advanced device modeling, even though it has veiled the intuitive nature of the modeling. More concretely, the adjustable tunnel mass pushes us to ignore the related issues that should carefully be considered. In this paper, we extract the tunnel masses for electrons and holes from an individual experiment and find that they are Formula Omitted, where Formula Omitted is the rest electron mass, irrespective of the molecular compound ratio between Formula Omitted and Formula Omitted and the film thickness. This suggests a convincing model for charge trapping in Formula Omitted including interfacial transition layers. It is also found that the leakage mechanism is the direct tunneling enhanced by the trapped positive charge.
The objective of this work is to evaluate various CO2 sequestration mechanisms occurring in the Morrow B Sandstone in the Farnsworth Unit. A history-matched numerical simulation model was created ...using extensive geological, petrophysical, and operational data collected from the field. The numerical model is competent to investigate the impact of residual, structural-stratigraphic, solubility, and mineral trapping mechanisms on the fluid transportation dynamics and petrophysical property variations. The model forecasts the field response of 20 years of WAG injections. Afterward, all wells were shut-in, and the reservoir was allowed to evolve for 1000 years to investigate the fate of injected CO2. In this paper, we assess the impacts of various trapping mechanisms on oil recovery and CO2 storage efficacy. By analyzing the results reported from the numerical simulation model, the in-situ fluid composition and mineralogy changes are also investigated. More importantly, we seek to confirm the petrophysical property variations due to the CO2 injection with observations from laboratory measurements. The experiences gained from this study provide valuable insights regarding physiochemical storage induced by the CO2 injection activities and serve as a benchmark case for future CO2 enhanced oil recovery (EOR) projects involving reactive solute transport.
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 an 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 <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula> 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, <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula>. Finding <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula>, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm. For a constant degree distribution and range of search, the worst case computational complexity of the proposed search algorithms for finding NETSs and stopping sets is linear in the code's block length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>. The average search complexity for stopping sets, however, is constant in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, if the simple cycles required as input to the search algorithm are available.
Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density ...parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth.
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