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。

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.