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

@article{Sziklai2013ASM,
  title={A small minimal blocking set in PG(n, pt), spanning a (t − 1)-space, is linear},
  author={P{\'e}ter Sziklai and Geertrui Van de Voorde},
  journal={Designs, Codes and Cryptography},
  year={2013},
  volume={68},
  pages={25-32}
}
In this paper, we show that a small minimal blocking set with exponent e in PG(n, pt), p prime, spanning a (t/e − 1)-dimensional space, is an $${\mathbb{F}_{p^e}}$$ -linear set, provided that p > 5(t/e)−11. As a corollary, we get that all small minimal blocking sets in PG(n, pt), p prime, p > 5t − 11, spanning a (t − 1)-dimensional space, are $${\mathbb{F}_p}$$ -linear, hence confirming the linearity conjecture for blocking sets in this particular case. 
4 Citations
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References

SHOWING 1-10 OF 16 REFERENCES
On 1-Blocking Sets in PG(n,q), n ≥ 3
TLDR
It is proved that in PG(n,q2),q = ph, p prime, p > 3,h ≥ 1, the second smallest minimal 1-blockingsets are the second largest minimal blocking sets, w.r.t. lines, in a plane of PG( n,q 2).
On the Linearity of Higher-Dimensional Blocking Sets
TLDR
To prove the linearity conjecture for $k$-blocking sets in $\mathrm{PG}(n,p^t)$, with exponent $e$ and $p^e\geq 7$ it is sufficient to prove it for one value of $n$ that is at least $2k$.
On multiple blocking sets in Galois planes
This article continues the study of multiple blocking sets in PG(2, q). In [A. Blokhuis, L. Storme, T. Szonyi, Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc.
Linear Point Sets and Rédei Type k-blocking Sets in PG(n, q)
In this paper, k-blocking sets in PG(n, q), being of Rédei type, are investigated. A standard method to construct Rédei type k-blocking sets in PG(n, q) is to construct a cone having as base a Rédei
Small Blocking Sets in Higher Dimensions
TLDR
It is shown that small blocking sets in PG(n, q) with respect to hyperplanes intersect every hyperplane in 1 modulo p points, where q=ph, which can be used to characterize certain non-degenerate blocking set in higher dimensions.
Blocking Sets in Desarguesian Affine and Projective Planes
In this paper we show that blocking sets of cardinality less than 3(q+ 1)/2 (q=pn) in Desarguesian projective planes intersect every line in 1 moduloppoints. It is also shown that the cardinality of
On the size of a blocking set inPG(2,p)
We show that the size of a non-trivial blocking set in the Desarguesian projective planePG(2,p), wherep is prime, is at least 3(p+1)/2. This settles a 25 year old conjecture of J. di Paola.
Proper blocking sets in projective spaces
  • U. Heim
  • Mathematics
    Discret. Math.
  • 1997
Congruences Involving Alternating Multiple Harmonic Sums
We show that for any prime $p\neq 2$, $$\sum_{k=1}^{p-1}{(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3}$$ by expressing the left-hand side as a combination of
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