Renitent lines

Abstract

There are many examples for point sets in finite geometry, which behave "almost regularly" in some (well-defined) sense, for instance they have "almost regular" line-intersection numbers. In this paper we investigate point sets of a desarguesian affine plane, for which there exist some (sometimes: many) parallel classes of lines, such that almost all lines of one parallel class intersect our set in the same number of points (possibly mod $p$, the characteristic). The lines with exceptional intersection numbers are called renitent, and we prove results on the (regular) behaviour of these renitent lines.

The sharpness of some statements has been proved. There are some new examples and a new section about some dual statements and their consequences regarding the geometric properties of codewords of the ${\mathbb{F}}_p$-linear code generated by characteristic vectors of lines of $\mathrm{PG}(2,q)$. The latter result generalise an old result of Blokhuis, Brouwer and Wilbrink

Cited by in corpus (1)

• On regular sets of affine type in finite Desarguesian planes and related codes