A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear

TL;DR

This paper proves that small minimal blocking sets in projective spaces over finite fields are linear under certain size and span conditions, confirming a conjecture in this specific case.

Contribution

It establishes conditions under which small minimal blocking sets are linear, advancing the understanding of their structure in finite projective spaces.

Findings

Small minimal blocking sets with certain parameters are F_p^e-linear.

All small minimal blocking sets in PG(n,p^t) with p>5t-11 are linear.

The linearity conjecture is confirmed for these cases.

Abstract

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case.