Theory of algebraic curves motivated by coding theory

这个项目下的每个结果都与厄米曲线有关。设q是素数p的幂，F是q^2个元素的有限域。厄米曲线是由非齐次方程y^q + y = x^<q+1> / f定义的平面曲线。这条曲线上的f -有序点的个数是关于格的最大值，这就是为什么人们在编码理论、有限几何等方面偏爱这条曲线的原因。之前我们已经确定了曲线上任何两点编码的最小权值的确切值。在这个项目中，我们试图找到厄米曲线上任意两点码的第二个汉明权重的精确值，我们相信我们已经成功了。另一个结果与可分态射的伽罗瓦理论有关。对于厄米曲线周围投影平面上的点P，我们考虑以P为中心的曲线的投影，证明了该投影形成伽罗瓦覆盖当且仅当该点是f有理的。此外，如果f有理点位于曲线上，则伽罗瓦群是Z/pZ的e个副本的直接和；若f有理点不在曲线上，则伽罗瓦群为Z/(q+1)。当点P不是f有理时，我们考虑投影的伽罗瓦闭包。我们已经找到了伽罗瓦群的场伽罗瓦闭包在场的目标线的投影。若该点不在曲线上，则伽罗瓦群是f线的投影一般线性群；如果曲线上的点是仿射f线的仿射一般线性群。

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.