Minimum size blocking sets of certain line sets with respect to an elliptic quadric in $PG(3,q)$
DOI:
https://doi.org/10.11575/cdm.v15i2.67943Abstract
For a given nonempty subset $\mathcal{L}$ of the line set of $\PG(3,q)$, a set $X$ of points of $\PG(3,q)$ is called an $\mathcal{L}$-blocking set if each line in $\mathcal{L}$ contains at least one point of $X$. Consider an elliptic quadric $Q^-(3,q)$ in $\PG(3,q)$. Let $\mathcal{E}$ (respectively, $\mathcal{T}, \mathcal{S}$) denote the set of all lines of $\PG(3,q)$ which meet $Q^-(3,q)$ in $0$ (respectively, $1,2$) points. In this paper, we characterize the minimum size $\mathcal{L}$-blocking sets in $\PG(3,q)$, where $\mathcal{L}$ is one of the line sets $\mathcal{S}$, $\mathcal{E}\cup \mathcal{S}$, and $\mathcal{T}\cup \mathcal{S}$.
References
A. Aguglia and M. Giulietti, Blocking sets of certain line sets related to a conic, Des. Codes Cryptogr. 39 (2006), 397--405.
A. Aguglia and G. Korchm'{a}ros, Blocking sets of external lines to a conic in $PG(2,q)$, $q$ odd, Combinatorica 26 (2006), 379--394.
A. Aguglia, G. Korchm'{a}ros and A. Siciliano, Minimal covering of all chords of a conic in $PG(2,q)$, $q$ even, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), 651--655.
A. Barlotti, Un'estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital. (3) 10 (1955), 498--506.
P. Biondi and P. M. Lo Re, On blocking sets of external lines to a hyperbolic quadric in $PG(3,q)$, $q$ even, J. Geom. 92 (2009), 23--27.
P. Biondi, P. M. Lo Re and L. Storme, On minimum size blocking sets of external lines to a quadric in $PG(3,q)$, Beitr"{a}ge Algebra Geom. 48 (2007), 209--215.
R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combinatorial Theory 1 (1966), 96--104.
B. De Bruyn, B. K. Sahoo and B. Sahu, Blocking sets of tangent and external lines to a hyperbolic quadric in $PG(3,q)$, Discrete Math. 341 (2018), 2820--2826.
B. De Bruyn, B. K. Sahoo and B. Sahu, Blocking sets of tangent lines to a hyperbolic quadric in $PG(3,3)$, Discrete Appl. Math. (2019), https://doi.org/10.1016/j.dam.2018.12.010.
M. Giulietti, Blocking sets of external lines to a conic in $PG(2,q)$, $q$ even, European J. Combin. 28 (2007), 36--42.
J. W. P. Hirschfeld, ``Finite projective spaces of three dimensions'', Oxford Mathematical Monographs, Oxford University Press, 1985.
R. Lidl and H. Niederreiter, ``Finite Fields'', Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, 1997.
G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. (3) 10 (1955), 507--513.
K. L. Patra, B. K. Sahoo and B. Sahu, Minimum size blocking sets of certain line sets related to a conic in $PG(2,q)$, Discrete Math. 339 (2016), 1716--1721.
B. K. Sahoo and B. Sahu, Blocking sets of tangent and external lines to a hyperbolic quadric in $PG(3,q)$, $q$ even, Proc. Indian Acad. Sci. Math. Sci. 129 (2019), no. 1, 129:4.
B. K. Sahoo and B. Sahu, Blocking sets of certain line sets to a hyperbolic quadric in $PG(3,q)$, Adv. Geom. (2019), https://doi.org/10.1515/advgeom-2018-0009.
B. K. Sahoo and N. S. N. Sastry, Binary codes of the symplectic generalized quadrangle of even order, Des. Codes Cryptogr. 79 (2016), 163--170.
B. Segre, On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arith. 5 (1959), 315--332.
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