1,721,062 research outputs found
Minimal blocking sets in PG(2,9)
No full textWe classify the minimal blocking sets of size 15 in P G(2, 9). We show
that the only examples are the projective triangle and the sporadic ex-
ample arising from the secants to the unique complete 6-arc in P G(2, 9).
This classification was used to solve the open problem of the existence
of maximal partial spreads of size 76 in P G(3, 9). No such maximal par-
tial spreads exist [13]. In [14], also the non-existence of maximal partial
spreads of size 75 in P G(3, 9) has been proven. So, the result presented
here contributes to the proof that the largest maximal partial spreads in
P G(3, q = 9) have size q 2 − q + 2 = 74
Small complete caps in spaces of even characteristic
No full textAbstractIn 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of order[formula]. Generalizing the construction of Segre, complete (qn+3(qn−1+…+q)+2)-caps inPG(2n, q),qeven,q⩾4, and complete (3(qn+…+q)+2)-caps inPG(2n+1, q),qeven,q⩾4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of order[formula]
Blocking sets in
No full textLet S be a Desarguesian (n-1)-spread of a hyperplane ∑ of PG(rn, q). Let Ω and B̄ be, respectively, an (n-2)-dimensional subspace of an element of S and a minimal blocking set of an ((r-1)n+1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base B̄ , and consider the point set B defined by
B=( K\∑ ) ∪ {X ∈ S : X ∩ K ≠ Ø}
in the Barlotti-Cofman representation of PG(r,qn) in PG(rn, q) associated to the (n-1)-spread S. Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61-81, 2006), under suitable assumptions on B̄ , we prove that B is a minimal blocking set in PG(r,qn). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size qn+2+1 in PG(r,qn), 3 ≤ r ≤ 6 and n ≥ 3, and of size q4+1 in PG(r,q2), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q3 Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q2+2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4,q2) of size q4 + 1, for any q a power of 3
A proof of the linearity conjecture for k-blocking sets in PG(n, p(3)), p prime
No full textIn this paper, we show that a small minimal -blocking set in \PG(n,q^3), , , prime, , intersecting every -space in points, is linear. As a corollary, this result shows that all small minimal -blocking sets in \PG(n,p^3), prime, , are -linear, proving the linearity conjecture (see \cite{sziklai}) in the case \PG(n,p^3), prime,
The use of blocking sets in Galois geometries and in related research areas
Full text linkBlocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
The Kakeya problem: a gap in the spectrum and classification of the smallest examples
No full textThe authors present a new example of a small Kakeya set in the affine plane AG(2,q) and they give the classification of the smallest Kakeya sets with at most 1/2q(q+2)+q/4 points, in case q even
On minimum size blocking sets of external lines to a quadric in PG(3,q)
No full textAbstract
We characterize the minimum size blocking sets with respect to the external
lines to a non-singular quadric or a quadric with a point vertex in
PG(d; q), d ≥ 4 and q ≥ 9. Our results show that these minimum size
blocking sets are equal to the sets of points not on the quadric in a suitably
chosen hyperplane with respect to the quadric
Minimal Blocking Sets in PG(2,8) and Maximal Partial Spreads in PG(3,8)
No full textWe prove that PG(2,8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3,8). This support the conjecture that q^2-q+2 is the largest size for a maximal partial spread of PG(3,q), q>7
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