As the demand for error-free data transmission and data storage increases, error control becomes increasingly important in data communication and data storage systems. It has become an integral part in almost every data communication or storage system design. Today very sophisticated error control schemes are being used in a broad range of communication and data storage systems to achieve reliable data transmission and storage, such as wireless communication, satellite communication, optical communication, hard disc drives, compact disks and many others. The objective of this research is to devise methods for constructing good error control codes and to develop efficient error control schemes which have great potential to achieve error-free communication and data storage for the future generation of data communication and storage systems.
Recently, there have been dramatic developments in error control codes and decoding algorithms. Two families of powerful codes, known as turbo and low density parity check (lDPC) codes, have been discovered and rediscovered. These two families of codes with iterative decoding have been shown to perform close to Shannon's theoretical limits with reasonable implementation complexity. As a result of their amazing error performance and practical implementation, it is anticipated that these two classes of codes will have an enormous impact on virtually all applications of error control coding over the next 10 years or so. This research involves in two important aspects of these two classes of Shannon limit approaching codes: construction of LDPC codes and turbo decoding of Reed-Solomon (RS) codes. The construction of LDPC codes is based on combinatoric appraches, such as finite geometries over finite fields, statistical experimental designs, permutation groups and graphs. In these approaches, points, lines, hyperplanes in finite geometries, balanced incomplete block designs, affine permutation groups, and paths and independent sets of graphs are used for constructing LDPC codes whose Tanner graphs do not contain short cycles. All the construction methods are systematic and codes constructed have good structural properties which simplify encoding and decoding implementations. Turbo decoding of a RS code is based on binary decomposition of the code into a set of simple binary component codes and formulating the code as a self concatenated code with the RS code itself as the outer code and the component codes as inner codes in a turbo coding arrangement. The decoding is carried out in two stages, turbo inner decoding and outer algebraic soft-decision decoding.
