In this paper, we show that a small minimal \$k\$-blocking set in \$\PG(n,q^3)\$, \$q=p^h\$, \$h\geq 1\$, \$p\$ prime, \$p\geq 7\$, intersecting every \$(n-k)\$-space in \$1\pmod{q}\$ points, is linear. As a corollary, this result shows that all small minimal \$k\$-blocking sets in \$\PG(n,p^3)\$, \$p\$ prime, \$p\geq 7\$, are \$\mathbb{F}_p\$-linear, proving the linearity conjecture (see \cite{sziklai}) in the case \$\PG(n,p^3)\$, \$p\$ prime, \$p\geq 7\$.

### A proof of the linearity conjecture for k-blocking sets in PG(n, p(3)), p prime

#### Abstract

In this paper, we show that a small minimal \$k\$-blocking set in \$\PG(n,q^3)\$, \$q=p^h\$, \$h\geq 1\$, \$p\$ prime, \$p\geq 7\$, intersecting every \$(n-k)\$-space in \$1\pmod{q}\$ points, is linear. As a corollary, this result shows that all small minimal \$k\$-blocking sets in \$\PG(n,p^3)\$, \$p\$ prime, \$p\geq 7\$, are \$\mathbb{F}_p\$-linear, proving the linearity conjecture (see \cite{sziklai}) in the case \$\PG(n,p^3)\$, \$p\$ prime, \$p\geq 7\$.
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2011
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11577/156738`
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