Theory of algebraic curves motivated by coding theory

受编码理论启发的代数曲线理论

基本信息

批准号：
17540045
负责人：
HOMMA Masaaki
金额：
$ 1.15万
依托单位：
Kanagawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

Each result under this project is concerned with a Hermitian curve. Let q be an e th power of a prime number p, and F the finite field of q^2 elements. A Hermitian curve is a plane curve defined by the inhomogeneous equation y^q + y = x^<q+1> over F. The number of F-rational points of this curve is the maximum value with respect the genus, which is a reason why people prefer this curve in constructing example in coding theory, in finite geometry etc.Previously we already determined the exact value of the minimum weight of any two-point code on the curve. Under this project, we tried to find exact value of the second Hamming weight of any two-point code on the Hermitian curve, and have succeeded we believe.The other result is related with Galois theory for a separable morphism. For a point P in the ambient projective plane of the Hermitian curve, we consider the projection from the curve with center P. We proved that the projection forms a Galois covering if and only if the point is F-rational. Moreover if the F-rational point lies on the curve, then the Galois group is the direct sum of e copies of Z/pZ ; if F-rational point does not lie on the curve, then the Galois group is Z/(q+1).When the point P is not F-rational, we consider the Galois closure of the projection. We have found out the Galois group for the field of the Galois closure over the field of the target line of the projection. If the point does not lie on the curve, the Galois group is the projective general linear group of an F-line ; if the point on the curve, it is the affine general linear group of an affine F-line.

该项目下的每个结果都涉及冬宫曲线。令Q为素数p的功率，而q的有限场元素元素的有限字段。 Hermitian曲线是由F的不均匀方程y^q + y = x^<q + 1>定义的平面曲线。该曲线的F理性点的数量是相对于属的最大值，这是一个人们喜欢在编码理论中构建示例，有限的几何形状等中构建示例的原因。我们已经确定了曲线上任何两点代码的最小重量的确切值。在这个项目下，我们试图在Hermitian曲线上找到任何两点代码的第二次锤击权重的确切价值，并已经成功地相信。另一个结果与Galois理论有关，以实现可分离的形态。对于Hermitian曲线的环境射击平面中的一个点P，我们考虑了CenterP的曲线的投影。我们证明，当且仅当点是F pational时，我们证明了该投影形成Galois覆盖率。此外，如果F理性点位于曲线上，则GALOIS组是Z/PZ的E副本的直接总和。如果F理性点不在曲线上，则GALOIS组为z/（q+1）。当点P不是F理性时，我们考虑投影的Galois闭合。我们已经发现了Galois组的Galois封闭场，这是投影范围的田地。如果该点不在曲线上，则Galois组是F线的投射通用线性组。如果曲线上的点是仿射F线的仿射通用线性组。