Department Colloquium

Location: ED 114

Speaker: Sam Adriaensen, Vrije Universiteit Brussel

Title:  Directions and codewords in finite planes

Abstract:

In this talk, we will take a look at affine and projective planes over finite fields. We will discuss two classical problems, and show how they are linked.

The first problem deals with directions determined by a point set in an affine plane. Set \(S\) is said to be equidistributed from direction \(d\) if all lines with slope \(d\) intersect \(S\) in the same number of points. Otherwise, we call direction \(d\) special. The natural question is to study sets with few special directions.

We will investigate a variation of this problem. We allow \(S\) to be a multiset, and we only look at the intersection of \(S\) with the lines modulo the characteristic of the field.

The second problem deals with the linear code \(C\) generated by the incidence matrix of the projective plane over the field Fq . In other words, we study the linear span of the characteristic functions of the lines in the plane, seen as subsets of the points. If \(q\) is not prime, it is known that the codewords of this code up to Hamming weight roughly \(q\)√\(q\) are just linear combinations of a small number of (characteristic vectors of) lines. However, if \(q\) is prime this is no longer the case, and there is an odd codeword of weight \(3(q − 1)\).

We will show a way to construct many more odd codewords of small weight for the case \(q\) prime, and we will show how this problem is very closely related to sets of points with few special directions in the modular sense.