1,125 research outputs found
The Finite Field Kakeya Problem
A Besicovitch set in AG(n,q) is a set of points containing a line in every
direction. The Kakeya problem is to determine the minimal size of such a set.
We solve the Kakeya problem in the plane, and substantially improve the known
bounds for n greater than 4.Comment: 13 page
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
Maximum Distance Separable Codes and Arcs in Projective Spaces
Given any linear code over a finite field we show how can be
described in a transparent and geometrical way by using the associated
Bruen-Silverman code. Then, specializing to the case of MDS codes we use our
new approach to offer improvements to the main results currently available
concerning MDS extensions of linear MDS codes. We also sharply limit the
possibilities for constructing long non-linear MDS codes.Comment: 18 Pages; co-author added; some results updated; references adde
Polynomials in finite geometry
Postprint (published version
- …