New Bounds for Linear Codes of Covering Radius 2

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

doi:10.1007/978-3-319-66278-7_1

Cited by 4 publications

(10 citation statements)

References 8 publications

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“…In light of this, we will adopt the following notation, which is widely used in the literature (e.g. [2,3,13,14]).…”

Section: Preliminariesmentioning

confidence: 99%

“…This is done by investigating the covering density µ q (n, r, C) of the code C, which equals the ratio of the total volume of these q n−r spheres to the volume of the space F n q (see e.g. [2,3,4,10,11,12]):…”

Section: Note Thatmentioning

confidence: 99%

“…In order to know which saturating sets are viewed as being 'small', we will be guided by the following lower bound on the size of arbitrary -saturating sets. Several variants of this bound were already known in the literature [2,3,4,12,13,14], but some only state the bound for specific values of , while others describe an approximate lower bound for large values of q. Proposition 4.2.1.…”

Section: Note Thatmentioning

confidence: 99%

“…Some results present upper bounds that are slightly larger than the desired one described in Open Problem 5.0.1. More specifically, the authors of articles [2,3,4,13,15], some with the aid of computer searches, present upper bounds on s q (N, ), ∈ {1, 2}, of the following form:…”

Section: N + 1 ≡ S +1mentioning

confidence: 99%

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Constructing saturating sets in projective spaces using subgeometries

Denaux

2021

Des. Codes Cryptogr.

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A -saturating set of PG(N, q) is a point set S such that any point of PG(N, q) lies in a subspace of dimension at most spanned by points of S. It is generally known that a -saturating set of PG(N, q) has size at least c • q N − +1 , with c > 1 3 a constant. Our main result is the discovery of a -saturating set of size roughly ( +1)( +2) 2 q N − +1 if q = (q ) +1 , with q an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of -saturating sets if < 2N −1 3 . As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes.To prove that this construction is a -saturating set, we observe that the affine parts of q -subgeometries of PG(N, q) having a hyperplane in common, behave as certain lines of AG + 1, (q ) N . More precisely, these affine lines are the lines of the linear representation of a q -subgeometry PG( , q ) embedded in PG + 1, (q ) N .

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“…In light of this, we will adopt the following notation, which is widely used in the literature (e.g. [2,3,13,14]).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Note Thatmentioning

confidence: 99%

Section: Note Thatmentioning

confidence: 99%

Section: N + 1 ≡ S +1mentioning

confidence: 99%

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Constructing saturating sets in projective spaces using subgeometries

Denaux

2021

Des. Codes Cryptogr.

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“…So, there is a one-to-one correspondence between [n, n − 4] q 3 codes and 2-saturating n-sets in PG (3, q). Therefore, q (4, 3) = s q (3,2).…”

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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