Theory of algebraic curves motivated by coding theory

该项目下的每个结果都与厄米特曲线有关。设q是素数p的e次幂，F是q^2个元素的有限域。埃尔米特曲线是由F上的非齐次方程y^q + y = x^<q+1>定义的平面曲线。这条曲线的F-有理点数是关于亏格的最大值，这也是人们在编码理论、有限几何等领域中构造例子时喜欢这条曲线的一个原因。以前我们已经确定了这条曲线上任意两点编码的最小权的精确值。在这个项目下，我们试图找到Hermitian曲线上任意两点码的第二Hamming重量的精确值，并且我们相信已经成功了。另一个结果与Galois关于可分态射理论有关。对于Hermitian曲线的环境射影平面上的点P，我们考虑中心为P的曲线的投影，证明了投影形成Galois覆盖当且仅当该点是F-有理数。若F-有理点在曲线上，则Galois群是Z/pZ的e个拷贝的直和;若F-有理点不在曲线上，则Galois群是Z/（q+1）;当点P不是F-有理点时，我们考虑投影的Galois闭包。我们求出了投影的目标线域上的伽罗瓦闭包域的伽罗瓦群。如果点不在曲线上，伽罗瓦群是F-线的射影一般线性群;如果点在曲线上，它是仿射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.