The primary objective of this research is the development of efficient techniques for reliable data transmission and storage. The specific areas addressed include (1) tighter lower bounds for the minimum distance of cyclic codes that can be easily computed and applied to Goppa codes and generalized Goppa codes (2) efficient decoding algorithms that can fully realize the error correcting capability of these codes,and (3) extension of promising approaches in decoding cyclic and Goppa Codes to improve the decoding of geometric Goppa codes. In the determination of tighter minimum distance bounds, investigations include root pattern characterization and codeword- locator polynomial approaches, and efficient methods of finding submatrices of the generalized Cauchy and generalized Vandermonde types for the associated Reed-Solomon codes. In decoding, the research includes a generalized Peterson procedure that can determine error locations from nonrecurrent dependence relations among the syndromes. A polynomial version of a fundamental iterative algorithm will be developed to provide a better alternative for hardware implementation. Extensions of the generalized Peterson procedure will be sought to decode the two subclasses of geometric Goppa codes, the elliptic codes, and the Hermitian codes, up to the actual minimum distance.
