Algebraic Low-Density Parity-Check Codes: Constructions, Trapping Sets, Corrections of Random Errors and Erasures and Transform Domain Approach

The ever-growing needs for cheaper, faster, and more reliable communication systems have forced many researchers to seek means to attain the ultimate limits on reliable communications. Low density parity-check (LDPC) codes are currently the most promising coding technique to achieve the Shannon capacities for a wide range of channels. Many LDPC codes have been chosen as the standard codes for various next generations of communication systems and they are appearing in recent data storage products. More applications are expected to come.;Major methods for constructing LDPC codes can be divided into two general categories, graph-theoretic-based methods (using computer search) and algebraic methods. Each type of constructions has its advantages and disadvantages in terms overall error performance, encoding and decoding implementations. In general, algebraically constructed LDPC codes have lower error-floor and their decoding using iterative message-passing algorithms converges at a much faster rate than the LDPC codes constructed using a graph theoretic-based method. Furthermore, it is much easier to construct algebraic LDPC codes with large minimum distances.;This research project is set up to investigate several important aspects of algebraic LDPC codes for the purpose of achieving overall good error performance required for future high-speed communication systems and high-density data storage systems. The subjects to be investigated include: (1) new constructions of algebraic LDPC codes based on finite geometries; (2) analysis of structural properties of algebraic LDPC codes, especially the trapping set structure that determines how low the error probability of a given LDPC code can achieve; (3) construction of algebraic LDPC codes and design coding techniques for correcting combinations of random errors and erasures that occur simultaneously in many physical communication and storage channels; and (4) analysis and construction of algebraic LDPC codes in transform domain.;Research effort has resulted in important findings in all four proposed research subjects which may have a significant impact on future generations of communication and storage systems and advance the state-of-the-art of channel coding theory.