Higgledy-Piggledy Sets in Projective Spaces of Small Dimension

Denaux, Lins

doi:10.37236/10736

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…Remark 5.7. Observe that a generalization of [7,Theorem 3.15] has been pointed out by Denaux in [17,Theorem 16]. This result states that a set of points {P 1 , .…”

Section: Characterization and Construction Of Outer Strong Blocking Setsmentioning

confidence: 92%

Outer Strong Blocking Sets

Alfarano¹,

Borello²,

Neri³

2023

Preprint

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Strong blocking sets, introduced first in 2011 in connection with saturating sets, have recently gained a lot of attention due to their correspondence with minimal codes. In this paper, we dig into the geometry of the concatenation method, introducing the concept of outer strong blocking sets and their coding theoretical counterpart. We investigate their structure and provide bounds on their size. As a byproduct, we improve the best-known upper bound on the minimum size of a strong blocking set. Finally, we present a geometric construction of small strong blocking sets, whose computational cost is significantly smaller than the previously known ones.

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“…Remark 5.7. Observe that a generalization of [7,Theorem 3.15] has been pointed out by Denaux in [17,Theorem 16]. This result states that a set of points {P 1 , .…”

Section: Characterization and Construction Of Outer Strong Blocking Setsmentioning

confidence: 92%

Outer Strong Blocking Sets

Alfarano¹,

Borello²,

Neri³

2023

Preprint

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“…Let q be a prime power. In the literature, the bound (1.1) is achieved in the following cases: 12,20,21,24]; R = sR , r = tR + s, q = (q ) R [12, 14]; r = tR, q is an arbitrary prime power [11,12,14,18,19].…”

Section: Introductionmentioning

confidence: 99%

“…In the literature, the bound (1.2) is achieved for special values of r, R, q: r = tR, q = (q ′ ) R [9,17,18,23]; R = sR ′ , r = tR + s, q = (q ′ ) R ′ [9, 10]; r = tR, q is an arbitrary prime power [9,10,14,15]; where t, s are integers and q ′ is a prime power.…”

Section: Introductionmentioning

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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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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“…Let q ′ be a prime power. In the literature, the bound (1.1) is achieved in the following cases: r ̸ = tR, q = (q ′ ) R [12,20,21,24]; R = sR ′ , r = tR + s, q = (q ′ ) R ′ [12,14]; r = tR, q is an arbitrary prime power [11,12,14,18,19].…”

mentioning

confidence: 99%