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

  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},
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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  • U. Heim
  • Mathematics
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  • 1997
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