Study on functions on finite fields and finite geometry

本文主要研究APN DHO（由APN函数构造）和双线性DHO。 给出了双线性DHO是APN DHO的一个充分必要条件。利用这个结果，我们表明，从一些APN功能的DHO的对偶和转置的对偶是未知的（到那时）。我们还给出了一个新的结构的Buratti-Del Fra DHO。利用这个构造，我们证明了Buratti-Del Fra DHO是一个双线性DHO，并且Buratti-Del Fra DHO的泛覆盖与Huybrechts DHO（APN DHO）的泛覆盖相同.本文还给出了Buratti-Del Fra DHO在PG（2d+1，2）中的商的一个例子。接下来，我们给出了PG（d（d+3）/2，2）中四个已知（全部已知）d维DHO（经典DHO）的统一描述.我们给出了PG（n，2）中n>2d+1的单连通d维DHO的一些例子。此外，我们还构造了PG（n，2）中许多新的对称双线性DHO，其中2d+1<n<d（d+3）/2.

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.