Theory of algebraic curves motivated by coding theory

这个项目下的每个结果都与一条厄米曲线有关。设q是素数p的e次方，F是q^2元的有限域。厄米曲线是由F上的非齐次方程y^q+y=x^&lt；q+1&gt所定义的一条平面曲线，它的F-有理点的个数是关于亏格的最大值，这也是人们在编码论、有限几何等领域中构造例子时偏爱这条曲线的原因。在这个项目下，我们尝试求出Hermitian曲线上任意两点编码的第二个Hamming重量的精确值，并且成功了。另一个结果与可分态射的Galois理论有关。对于Hermitian曲线的环境射影平面上的一点P，我们考虑从以P为中心的曲线的投影，证明了该投影形成Galois覆盖的充要条件是该点是F-有理的。此外，如果F-有理点位于曲线上，则Galois群是Z/PZ的e个拷贝的直和；如果F-有理点不在曲线上，则Galois群是Z/(q+1).当点P不是F-有理时，我们考虑投影的Galois闭包.我们已经找出了伽罗瓦群对于场的伽罗瓦闭合在场的目标线上的投影。如果点不在曲线上，则Galois群是F-直线的射影广义线性群；如果点在曲线上，则它是仿射F-直线的仿射广义线性群。

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.