Transitive $$\text{ PSL}(2,7)$$ -invariant 42-arcs in 3-dimensional projective spaces

Sonnino, Angelo

doi:10.1007/s10623-012-9778-z

Cited by 9 publications

(7 citation statements)

References 5 publications

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“…We denote by t(3, q) the size of the smallest known complete arc in PG(3, q). The arcs obtained in [6,20,21,23] and in this paper (one arc of [54] is used also) give the value of t(3, q) providing the following theorem, cf. (2.2).…”

Section: We Call the Value Cmentioning

confidence: 81%

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: 81%

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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“…For , the FOP and 2FOP algorithms are applied, in preference, but for , we used the d -Rand-Greedy algorithm. For , the complete 42-arc of [ 25 ] is taken. …”

Section: The Smallest Known Complete Arcs In Pg (3mentioning

confidence: 99%

Bounds for Complete Arcs in $$\mathrm {PG}(3,q)$$ and Covering Codes of Radius 3, Codimension 4, Under a Certain Probabilistic Conjecture

Davydov

Marcugini

Pambianco

2020

Lecture Notes in Computer Science

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Let t ( N , q ) be the smallest size of a complete arc in the N -dimensional projective space over the Galois field of order q . The d -length function is the smallest length of a q -ary linear code of codimension (redundancy) r , covering radius R , and minimum distance d ; in particular, is the smallest length n of an quasi-perfect MDS code. By the definitions, . In this paper, a step-by-step construction of complete arcs in is considered. It is proved that uncovered points are uniformly distributed in the space. A natural conjecture on quantitative estimations of the construction is presented. Under this conjecture, new upper bounds on t (3, q ) are obtained, in particular, .

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“…This approach produced several examples of small PD-sets that, in some cases, can be proven to be the smallest possible. In general, it is convenient to start with a linear code with a large prescribed automorphism group and, not only because of permutation decoding, these codes have been widely investigated in the last years, see [2,4,6,9,8,10,11,12,13,14,18,21,24,36,37,38,39,40,41,49,50,51,52,54,56] and references therein. This paper deals with Problem (1).…”

Section: Introductionmentioning

confidence: 99%

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

Pace¹,

Sonnino²

2021

Advances in Mathematics of Communications

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This paper deals with the problem of determining whether a PDset exists for a given linear code C and information set I. A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups M 12 and M 22 , respectively. In both cases, the existence of a PD-set is proven. In general, the algorithm works well whenever the code C has a very large automorphism group.

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Transitive $$\text{ PSL}(2,7)$$ -invariant 42-arcs in 3-dimensional projective spaces

Cited by 9 publications

References 5 publications

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

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

Bounds for Complete Arcs in $$\mathrm {PG}(3,q)$$ and Covering Codes of Radius 3, Codimension 4, Under a Certain Probabilistic Conjecture

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

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