Generalized Hamming Weights of Algebraic-Geometric Codes and Asmptotically Good Algebraic-Geometric Codes

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ABSTRACT: NCR-9505619, University of Southwestern Louisiana, Generalized Hamming Weights of Algebraic-Geometric Codes and Asymptotically Good Algebraic-Geometric Codes, PI-Gui-Liang Feng, and T.R.N. Rao: This research will explore the following problems: (1) to develop a simple and efficient method to determine a lower bound of the generalized Hamming weights of algebraic-geometric (AG) codes, such that the method can also be used to determine the weight hierarchy of a special class of AG codes, for example, Hermitian codes and generalized Hermitian codes, and (2) to investigate the construction of asymptotically good improved AG codes, and improved AG codes that exceed the Gilbert-Varshamov bound for some special finite fields. A more general approach will be developed to determine the minimum distance bound, which is as good or better than the bound given by the Reiman-Roch theorem (referred to here as the Goppa Bound) or by counting the number of well behaving terms (known as the Feng-Rao bound). The Feng-Rao bound was shown to be superior to the Goppa bound. Based on this approach the principal investigators will develop a lower bound of the generalized Hamming weights. For special classes of AG codes, this method can be used to determine the weight hierarchy. Classes of AG codes with true minimum distance greater than the Goppa and the Feng-Rao bounds will be constructed which provide good code properties for length and minimum distance. An attempt will be made to prove that these codes are asymptotica lly good codes. The problem of finding a sequence of AG codes exceeding the Gilbert-Varshamov bound will be the main feature of the research.***************************************************************************** Aubrey M. BushProgram Director, Acting Deputy Divison DirectorDivision of Networking and Communications Research and InfrastructureNational Science Foundation