Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

LDPC codes from voltage graphs

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained

EXTREMAL ABSORBING SETS IN LOW-DENSITY PARITY-CHECK CODES

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms

Chapter Three. EED library as a basis for systematic reviews

3.1 Defining Systematic Review Question Priorities 3.2 Determining Relevance to the Systematic Review 3.3 Acquisition of References and Copyright Fair Use Compliance 3.4 Documenting Relevance to the Systematic Review 3.5 Data Extraction for the Systematic Review 3.6 EED Library: Search Results Overview 3.7 Quality Control 3.8 EED Library Statushttps://digitalcommons.wustl.edu/tropicalenteropathybook/1004/thumbnail.jp

Chapter Five. Systematic review results by biomarker classifications

5.1 Markers of Absorption and Permeability Overview 5.2 Markers of Absorption 5.3 Markers of Permeability 5.4 Markers of Digestion 5.5 Markers of Intestinal Inflammation and Intestinal Immune Activation 5.6 Markers of Systemic Inflammation and Systemic Immune Activation 5.7 Markers of Microbial Drivers 5.8 Markers of Nonspecific Intestinal Injury 5.9 Markers of Extra-Small Intestinal Function 5.10 Relationships Between Markers of EED, Including Histopathology 5.11 Relationships between EED Biomarkers and Growth or Other Outcomes of Interesthttps://digitalcommons.wustl.edu/tropicalenteropathybook/1006/thumbnail.jp