Study on functions on finite fields and finite geometry

我们主要研究了APN DHO(由APN函数构造)和双线性DHO。给出了双线性DHO为APN DHO的充要条件。利用这一结果，我们证明了某些APN函数的DHO的对偶和对偶的转置是未知的。我们还给出了一种新的Buratti-Del Fra DHO结构。利用这种构造，我们证明了Buratti-Del Fra DHO是双线性DHO，并且Buratti-Del Fra DHO的泛覆盖与Huybrechts DHO(APN DHO)的泛覆盖相同。我们还给出了Pg(2d+1，2)中的Buratti-Del Fra DHO的商的一个例子。接下来，我们给出了PG(d(d+3)/2，2)(经典DHO)中四个已知的(都是已知的)d维DHO的统一描述。给出了PG(n，2)中具有n&gt；2d+1的单连通d维DHO的例子，并构造了许多新的对称双线性DHO。

We mainly study on APN DHO(constructed from APN functions) and bilinear DHO. We give a necessary and sufficient condition for a bilinear DHO to be an APN DHO. Using this result, we show that the dual and the transpose of the dual of the DHOs from some APN function are not known (by that time). We also give a new construction of the Buratti-Del Fra DHO. Using this construction, we show that the Buratti-Del Fra DHO is a bilinear DHO, and that the universal cover of the Buratti-Del Fra DHO is same as that of the Huybrechts DHO (APN DHO). We also give an example of a quotient of the Buratti-Del Fra DHO in PG(2d+1,2). Next, we give an uniform description for four known (all known) d-dimensional DHOs in PG(d(d+3)/2,2)(classical DHOs). We give some examples of simply connected d-dimensional DHOs in PG(n,2) with n>2d+1. Moreover, we construct many new symmetric bilinear DHOs in PG(n,2) for 2d+1<n<d(d+3)/2.