On The Construction of Long and Efficient Algebraic Geometric Codes

长而高效的代数几何代码的构造

• 批准号: 9714626
• 负责人: P. Vijay Kumar
• 金额: $ 8.5万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-05-15 至 2002-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9714626&HistoricalAwards=false
• 关键词: Construction; Long; Efficient; Algebraic; Geometric

Around 1980, V. D. Goppa used the theory of algebraic curves to construct a new family of codes, now referred to as algebraic geometric (AG) codes. Code performance depended upon certain curve parameters. Shortly after, Tsfasman, Vladut and Zink showed the existence of curves such that the resulting AG codes had long lengths and performance exceeding that of the Gilbert-Varshamov (G-V) bound -- a feat that until then was widely considered unattainable. However, the Tsfasman-Vladut-Zink result does not provide an explicit description of the curves. Recently, Garcia and Stichtenoth exhibited two families of curves having an explicit and simple description, that also give rise to codes better than the G-V bound. An explicit description of the AG code is still lacking however, as this requires the determination of a basis for a certain vector space, consisting of functions defined on the curve. This remains an open problem and its solution using the theory of function fields is a principal goal of the research activity under this grant. Recent progress in decoding AG codes has also increased interest in this problem. Also under investigation are a second set of curves arising from the trace function over Galois rings and the intersection of hyperplanes in projective space. The latter have application to the study of codes and pseudonoise sequences.

1980年左右，V。D. GOPPA使用代数曲线理论来构建一个新的代码家族，现在称为代数几何（AG）代码。 代码性能取决于某些曲线参数。 此后不久，TSFASMAN，VLADUT和ZINK显示出曲线的存在，因此所得的AG代码的长度和性能超过了Gilbert-Varshamov（G-V）的结合，这一壮举被广泛认为是无法实现的。 但是，TSFASMAN-VLADUT-ZINK结果并未提供曲线的明确描述。 最近，加西亚（Garcia）和斯蒂奇·斯蒂诺斯（Stichtenoth）展示了两个具有明确和简单描述的曲线家族，这些曲线也比G-V绑定更好地产生代码。 但是，仍然缺乏对AG代码的明确描述，因为这需要确定特定向量空间的基础，该矢量空间由曲线定义的函数组成。 这仍然是一个开放的问题，其使用功能领域理论的解决方案是该赠款下研究活动的主要目标。 解码AG代码的最新进展也增加了对此问题的兴趣。 还在研究的是第二组曲线，该曲线是由Galois环上的痕量功能和投射空间中超平面的交汇处引起的。 后者有针对代码和伪序列的研究。