基于Kummer扩张的代数几何码的若干问题研究

Algebraic geometric codes have widely applications in areas such as cryptography and communications. Recently, a lot of researches are dedicated to the algebraic geometric codes arising from Hermitian curves. In this proposal, we focus our attention on the properties of algebraic geometric codes over Kummer extensions, extending the results of the corresponding Hermitian codes. The details are described as follows:. 1) generalize these results of gaps and pure gaps of a pair of points over Hermitian curves, compute the numbers of Weierstrass gaps and pure gaps with several points over Kummer extensions, and construct optimal algebraic geometric codes over Kummer extensions;.2) generalize these results of order bound of Hermitian two-point codes, give efficient algorithms of order bound and generalized order bound associated with one- and two-point codes over Kummer extensions, respectively, and consider cases of multi-point codes and construct optimal codes achiving some well-known bounds;.3) generalize these results of Hamming weights of Hermitian two-point codes and general one-point algebraic geometric codes, estimate precisely the Hamming weights of Hermitian three-point (collinear) codes, two-point and three-point (collinear) codes over Kummer extensions, and construct new optimal locally recoverable codes over Kummer extensions achieving some well-known bounds..All the above are basic problems in coding theory and they will have many applications in theory and practice in the future.

代数几何码在密码学和通信中有广泛应用。目前Hermitian码的研究结果很多。本项目拟推广Hermitian码的部分理论，研究基于Kummer扩张的代数几何码的若干问题，具体地：.1) 推广Hermitian两点间隙和纯间隙的结果，计算 Kummer扩张多点Weierstrass间隙和纯间隙的个数，构造最优Kummer扩张码；.2) 推广Hermitian两点码order界的结果，研究Kummer扩张单点码和两点码的order界和广义order界的有效计算方法，考虑多点码的情形，构造达到已知界的最优码；.3) 推广Hermitian两点码和一般单点代数几何码广义汉明重量的结果，给出Hermitian (共线的)三点码、Kummer扩张两点码和(共线的)三点码的广义汉明重量的精确估计，构造新的基于Kummer扩张的最优局部恢复码。.以上问题是编码理论的基本内容，具有较好的理论价值和应用前景。

本项目(11701317)主要研究基于Kummer扩张的代数几何码的若干问题。.1.研究一类特殊的Kummer扩张,即Hermitian商曲线上多点的Weierstrass间隙和纯间隙,解决了纯间隙的计数问题,并给出两点情况下间隙和纯间隙的计数结果,对Hermitian曲线上两点的纯间隙计数理论进行合理的推广。.2.研究基于GGS曲线的多点的代数几何码的参数问题,这是对GK曲线上相应结果的必要的推广,并构造了参数优于MinT码表的记录的多点码。.3.构造了几类新的特殊的线性码,寻找最优码和几乎最优码,研究它们的参数以及它们的完全重量分布,这些码都具有较少的重量,它们是二重码或者三重码,且都是极小码,可用于构建秘密共享方案。.4.研究一些特殊的代数几何码的参数问题,即LCD BCH码的参数和LCD MDS码的参数,并提出了新的构造LCD MDS码方法。.5.在研究代数几何码的同时,我们发现量子码的一些有趣的结果。我们研究二元和非二元的非对称量子码的重量分布和最小距离的上界问题,成功解决了“非对称量子码满足Singleton界”的公开问题,为构造量子MDS码建立了理论基础。