代数几何码的改进列表译码

Algebraic curves over finite fields have been widely applied in many fields of information science since late last century. One important invention is the algebraic geometric codes for error correction. It is a deep application of mathematical method to explicitly constructing algebraic geometric codes from algebraic curves over finite fields. In recent ten years by choosing specific divisors on certain algebraic curves many good results on the construction of algebraic geometric codes and their bounds have been obtained by many researchers including the applicant. In this project, we aim at extending the list decoding method of correlated Reed-Solomon codes and folded Reed-Solomon codes to the improved list decoding of algebraic geometric codes. We also put an effort to reduce the complexity of the decoding algorithm for the implementation of algebraic geometric codes efficiently for real application.

有限域上代数曲线理论自上世纪后半叶以来在信息科学的许多领域得到了应用,其中一个重要应用是发明了应用于纠错的代数几何码。通过具体的有限域和其上的代数曲线明确构造出的代数几何码揭示了一种深刻的数学方法的应用，出现不久就使纠错码理论得到重要理论突破。最近十年，通过具体的代数曲线和其上除子的选取，国际上包括申请人在内在代数几何码的构造和界的研究上获得了诸多好的结果。本项目研究代数几何码的改进列表译码方法。目标是将关联Reed-Solomon码，折叠Reed-Solomon码的列表译码方法推广到代数几何码的译码上并改进列表译码算法的复杂度。为代数几何码的真正实际应用找到有效的实现方法。

基于代数几何码的秘密共享方案在2006美密会上首次提出。这种方案是所谓ramp方案，其准入集和禁止集间有大小为2g的未知情形，其中g是所依赖的代数曲线的亏格.我们把对应椭圆曲线的线性秘密共享方案的构造推广到任意亏格的超椭圆曲线上，并且对理想的和带权重的超椭圆线性秘密共享方案都把2g 的未知情形减少到g-1。我们研究了有限域上分圆多项式的明确分解式。我们证明了在一个额外条件下,我们可以分解更多的分圆多项式,诸如次数为3^n, 3^n.5 和3^n.7。此外本课题还研究了椭圆曲线密码所涉及的椭圆曲线的点乘运算以及高次扭(Twist)双线性对的计算等问题。