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

摘要：NCR-9505619，西南路易斯安那大学，代数-几何码和渐近良好代数-几何码的广义Hamming权，冯贵良，饶仁宁：本研究将探讨以下问题：(1)开发的一种简单而有效的方法来确定一个下界的广义汉明重量algebraic-geometric (AG)代码,这样的方法也可以用来确定一个特殊类的重量等级的AG)代码,例如,埃尔米特代码和广义厄密码,(2)探讨建设和渐近好AG)编码,改善和提高AG)编码超过Gilbert-Varshamov运往一些特殊的有限的领域。我们将开发一种更一般的方法来确定最小距离界，它与Reiman-Roch定理（这里称为Goppa界）给出的界一样好，甚至更好，或者通过计算表现良好的项的数量（称为Feng-Rao界）。丰饶系优于哥帕系。基于这种方法，主要研究者将开发一个广义汉明权值的下界。对于特殊类别的AG代码，该方法可用于确定权重层次。本文将构造具有比Goppa界和Feng-Rao界更大的真最小距离的AG码类，它们具有较好的码长和最小距离特性。我们将尝试证明这些码是渐近的好码。寻找超过Gilbert-Varshamov界的AG编码序列的问题将是本研究的主要特点。***************************************************************************** 奥布里·m·BushProgram导演,代理副分工和DirectorDivision网络和通信研究和InfrastructureNational科学基金会

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