On the linearity of higher-dimensional blocking sets

TL;DR

This paper investigates the linearity conjecture for small minimal k-blocking sets in projective spaces, establishing reduction techniques that relate the conjecture's validity across different dimensions and field sizes.

Contribution

It demonstrates that proving linearity in a specific dimension suffices to establish it in higher dimensions for certain parameters, advancing understanding of the conjecture's scope.

Findings

Linearity in PG(2, q) implies linearity in higher dimensions under certain conditions.

Reducing the problem to a single dimension simplifies the proof of the linearity conjecture.

The paper extends the linearity conjecture's implications to larger projective spaces.

Abstract

A small minimal k-blocking set B in PG(n, q), q = pt, p prime, is a set of less than 3(qk + 1)/2 points in PG(n, q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies this property. The linearity conjecture states that all small minimal k-blocking sets in PG(n, q) are linear over a subfield Fpe of Fq. Apart from a few cases, this conjecture is still open. In this paper, we show that to prove the linearity conjecture for k- blocking sets in PG(n, pt), with exponent e and pe \geq 7, it is sufficient to prove it for one value of n that is at least 2k. Furthermore, we show that the linearity of small minimal blocking sets in PG(2, q) implies the linearity of small minimal k-blocking sets in PG(n, pt), with exponent e, with pe \geq t/e + 11.