LDPC Code Generation Using Hamiltonian Bipartite Graph Catalogs
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Solution Overview
Problem
Conventional methods for generating error-correction codes face difficulties in finding trivalent graphs of large girth, particularly outside the vertex-transitive class, and struggle to provide a sufficient supply of non-isomorphic graphs for orders greater than 32, limiting their effectiveness in wireless communication systems.
Innovation Solution
A system and method for generating a catalog of (3, g) Hamiltonian bipartite graphs, which are used to create a catalog of Low-Density Parity Check (LDPC) codes, including lists of non-isomorphic trivalent graphs for orders greater than 32, and these graphs are used to generate error-correction codes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional methods (LDPC codes, Turbo Codes) are used for generating error-correction codes, then error detection and correction capabilities are provided, but difficulty arises in finding trivalent graphs of large girth beyond vertex-transitive class
Solution Approach 1:
The patent segments the graph generation process by focusing specifically on Hamiltonian bipartite graphs with girth 16, dividing the complex problem of finding all trivalent graphs into a manageable subset with specific properties that can be systematically constructed and cataloged
Solution Approach 2:
The patent changes the parameters of the graph structure by specifying exact constraints (3-regular, Hamiltonian, bipartite, girth 16) to transform the general problem of finding error-correction graphs into a specific constructive problem with known solutions
2Reliability
If conventional methods are used, then error-correction codes can be generated, but insufficient supply of non-isomorphic graphs for orders greater than 32 is available
Solution Approach 1:
The patent performs preliminary action by pre-generating and cataloging 184 non-isomorphic Hamiltonian bipartite graphs with girth 16 before they are needed for error-correction applications, making them immediately available for use in communication systems
Solution Approach 2:
The patent creates multiple copies of graph structures through systematic construction methods, generating 184 distinct non-isomorphic graphs that can be used as templates for creating error-correction codes with various parameters
3Reliability
If vertex-transitive graphs are used, then some error-correction codes can be generated, but the supply is limited and does not extend to orders greater than 32
Solution Approach 1:
The patent moves away from the symmetric vertex-transitive graphs to asymmetric Hamiltonian bipartite graphs, breaking the symmetry constraint to access a much larger family of graphs with orders greater than 32 while maintaining the required error-correction properties
Solution Approach 2:
The patent adds the dimension of bipartite structure to the graph construction, transforming the problem from finding general trivalent graphs to finding 3-regular Hamiltonian bipartite graphs, which opens up new families of graphs with different order ranges
Data Source
AI summary
The present invention provide a system and method for generating a catalogue of graphs that acts as a source for error correcting codes. A D3 chord index notation is used to describe the graphs. A list of (3, g) Hamiltonian graphs for even girth g is created to satisfy the condition 6 g 16. Each of the lists is infinite and is used for creating LDPC codes of high quality. Furthermore, the embodiments herein generalizes the application scenario for usage of LDPC codes. The embodiments herein utilizes dynamically changing error-correction codes in wireless communication systems.