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. 
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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…
On the code generated by the incidence matrix of points and k-spaces in PG(n, q) and its dual
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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