New upper bounds on the smallest size of a saturating set in a projective plane

Bartoli, Daniele; Davydov, Alexander A.; Giulietti, Massimo; Marcugini, Stefano; Pambianco, Fernanda

doi:10.1109/red.2016.7779320

Cited by 5 publications

(5 citation statements)

References 24 publications

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“…In Figure 1, the sizes t(3, q) of the smallest known [6,20,21,23,54] complete arcs in PG(3, q) divided by 3 √ q ln q are shown by the bottom curve. The upper bounds of Theorem 6.10 and Theorem 3.3 for R = 3, t = 1 (also divided by 3 √ q ln q) are given by the top curve; the value λ = 3 is used, see Table 1.…”

Section: We Call the Value Cmentioning

confidence: 99%

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Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

Davydov¹,

Marcugini²,

Pambianco³

2023

AMC

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The length function <inline-formula><tex-math id="M3">\begin{document}$ \ell_q(r,R) $\end{document}</tex-math></inline-formula> is the smallest length of a <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary linear code with codimension (redundancy) <inline-formula><tex-math id="M5">\begin{document}$ r $\end{document}</tex-math></inline-formula> and covering radius <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In this work, new upper bounds on <inline-formula><tex-math id="M7">\begin{document}$ \ell_q(tR+1,R) $\end{document}</tex-math></inline-formula> are obtained in the following forms:<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} &(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} &(b)\; \ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula>In the literature, for <inline-formula><tex-math id="M8">\begin{document}$ q = (q')^R $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ q' $\end{document}</tex-math></inline-formula> a prime power, smaller upper bounds are known; however, when <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> is an arbitrary prime power, the bounds of this paper are better than the known ones.For <inline-formula><tex-math id="M11">\begin{document}$ t = 1 $\end{document}</tex-math></inline-formula>, we use a one-to-one correspondence between <inline-formula><tex-math id="M12">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes and <inline-formula><tex-math id="M13">\begin{document}$ (R-1) $\end{document}</tex-math></inline-formula>-saturating <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>-sets in the projective space <inline-formula><tex-math id="M15">\begin{document}$ \mathrm{PG}(R,q) $\end{document}</tex-math></inline-formula>. A new construction of such saturating sets providing sets of small size is proposed. Then the <inline-formula><tex-math id="M16">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "<inline-formula><tex-math id="M17">\begin{document}$ q^m $\end{document}</tex-math></inline-formula>-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension <inline-formula><tex-math id="M18">\begin{document}$ r = tR+1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}$ t\ge1 $\end{document}</tex-math></inline-formula>.

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Section: We Call the Value Cmentioning

confidence: 99%

“…Figure1. The sizes t(3, q) of the smallest known complete arcs in PG(3, q) (bottom curve) vs upper bounds of Theorem 6.10 and Theorem 3.3 (top curve) for R = 3, t = 1; the sizes and bounds are divided by3 √ q ln q; λ = 3…”

mentioning

confidence: 99%

Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

Davydov¹,

Marcugini²,

Pambianco³

2023

AMC

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“…Multiple saturating sets (see e.g. in [3]) and their generalizations in higher dimensional spaces are also investigated. A point set S in PG(n, q) is saturating if any point of PG(n, q) \ S is collinear with two points in S. The two proof techniques presented in Sections 2 and 3 are applicable in these more general settings as well.…”

Section: Connections With Hypergraph Coverings Applications and Openmentioning

confidence: 99%

“…However, in the general case when the plane is not necessarily Desarguesian or the order is arbitrary, we only have much weaker results. Following the footprints of Boros, Szőnyi and Tichler [6], Bartoli, Davydov, Giulietti, Marcugini and Pambianco [3] obtained an estimate on the minimal size of a saturating set in Π q . Proposition 1.5 (Bartoli et al).…”

Section: Introductionmentioning

confidence: 98%

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Saturating sets in projective planes and hypergraph covers

Nagy

2018

Discrete Mathematics

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Let Π q be an arbitrary finite projective plane of order q. A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈ √ 3q ln q⌉ + ⌈( √ q + 1)/2⌉. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.

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