Applications of finite fields to combinatorics and finite geometries
This paper on applications of finite fields to combinatorics and finite geometry is in three parts. In the first part the author obtains a bound on the degree of a polynomial \(f\in F[x_1,\dots,x_n]\), \(F= \text{GF}(q)\) satisfying some local conditions with respect to its action on the points of the \(n\)-dimensional affine space AG\((n, F)\). The result yields an interesting application on a \(t\)-intersection set \(W\) in AG\((n, F)\), which is a set of points in \(\Omega= \text{AG}(n, F)\), such that each hyperplane in \(\Omega\) contains at least \(t> 0\) points of \(W\). By using the bound, such a subset of the affine space contains at least \((n+ t- 1)(q- 1)+1\) points. The second part of the paper is devoted to the analysis of the \(p\)-rank of an incidence matrix of a finite linear space with respect to the prime field GF\((p)\). The author gives an important lower bound of the \(p\)-rank of such a matrix. Especially, his result says that the \(p\)-rank of a finite projective plane is at least \(n\sqrt{n}+ 1\), where \(p\) is a prime dividing the order \(n\) of the plane. Finally, the last part deals with certain MDS codes and BRUCK nets. In particular, the author constructs a class of so-called unimbeddable nets.
scientific article; zbMATH DE number 5070968