Codes with few weights arising from linear sets

Napolitano, Vito; Zullo, Ferdinando

doi:10.3934/amc.2020129

Cited by 9 publications

(5 citation statements)

References 33 publications

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“…As a consequence of Theorem 7.16 and the connection between linear rank metric codes and q-systems (see [28] and also [1]), the linear rank metric codes having as a generator matrix G have exactly three nonzero weights, which are n − r, n − 1, n. In particular, they are examples (r − 1)-almost MRD codes, see [12]. Moreover, using [1, Theorem 4.8], we can from L U we can also construct linear Hamming metric codes with only three weight and for which we can completely establish its weight distribution, as already done for some classes of linear sets (see also [25,33]).…”

Section: More Preciselymentioning

confidence: 87%

Multi-Sidon spaces over finite fields

Zullo¹

2021

Preprint

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Sidon spaces have been introduced by Bachoc, Serra and Zémor in 2017 in connection with the linear analogue of Vosper's Theorem. In this paper, we propose a generalization of this notion to sets of subspaces, which we call multi-Sidon space. We analyze their structures, provide examples and introduce a notion of equivalnce among them. Making use of these results, we study a class of linear sets in PG(r − 1, q n ) determined by r points and we investigate multi-orbit cyclic subspace codes.

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Section: More Preciselymentioning

confidence: 87%

Multi-Sidon spaces over finite fields

Zullo¹

2021

Preprint

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“…Remark 5.10. In [75,93], the distribution of the intersection between an s-scattered F q -subspace of dimension km/(s + 1) and the hyperplanes of V have been determined.…”

Section: Intersection With Hyperplanesmentioning

confidence: 99%

On subspace designs

Santonastaso¹,

Zullo²

2022

Preprint

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The aim of this paper is to investigate the theory of subspace designs, which have been originally introduced by Guruswami and Xing in 2013 to give the first construction of positive rate rank-metric codes list decodable beyond half the distance. However, sets of subspaces with special pattern of intersections with other subspaces have been already studied, such as spreads, generalised arcs and caps, and Cameron-Liebler sets. In this paper we provide bounds involving the dimension of the subspaces forming a subspace design and the parameters of the ambient space, showing constructions satisfying the equality in such bounds. Then we also introduce two dualities relations among them. Among subspace designs, those that we call s-designs are central in this paper, as they generalize the notion of s-scattered subspace to subspace design and for their special properties. Indeed, we prove that, for certain values of s, they correspond to the optimal subspace designs, that is those subspace designs that are associated with linear maximum sum-rank metric codes. Special attention has been paid for the case s = 1 for which we provide several examples, yielding surprising to families of two-intersection sets with respect to hyperplanes (and hence two-weight linear codes). Moreover, s-designs can be used to construct explicit lossless dimension expanders (a linear-algebraic analogue of expander graphs), without any restriction on the order of the field. Another class of subspace designs we study is those of cutting designs, since they extend the notion of cutting blocking set recently introduced by Bonini and Borello. These designs turn out to be very interesting as they in one-to-one correspondence with minimal sum-rank metric codes. The latter codes have been introduced in this paper and they naturally extend the notions of minimal codes in both Hamming and rank metrics.

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“…Our aim is to find codes arising from linear sets (or pointsets related to them) having few weights in the spirit of [30,38], see also [1,Section 4]. Let PG(2, q n ) = PG(V, F q n ), with dim F q n (V ) = 3, up to coordinatize, we can suppose that V = F 3 q n .…”

Section: Introductionmentioning

confidence: 99%

Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2023

AMC

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In this paper we consider two pointsets in <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula> arising from a linear set <inline-formula><tex-math id="M3">\begin{document}$ L $\end{document}</tex-math></inline-formula> of rank <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> contained in a line of <inline-formula><tex-math id="M5">\begin{document}$ \mathrm{PG}(2,q^n) $\end{document}</tex-math></inline-formula>: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set <inline-formula><tex-math id="M6">\begin{document}$ L $\end{document}</tex-math></inline-formula>. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing <inline-formula><tex-math id="M7">\begin{document}$ L $\end{document}</tex-math></inline-formula> to be an <inline-formula><tex-math id="M8">\begin{document}$ {\mathbb F}_{q} $\end{document}</tex-math></inline-formula>-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the <inline-formula><tex-math id="M9">\begin{document}$ \Gamma\mathrm{L} $\end{document}</tex-math></inline-formula>-class of <inline-formula><tex-math id="M10">\begin{document}$ L $\end{document}</tex-math></inline-formula> and the number of inequivalent codes we can construct starting from it.

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Codes with few weights arising from linear sets

Cited by 9 publications

References 33 publications

Multi-Sidon spaces over finite fields

Multi-Sidon spaces over finite fields

On subspace designs

Two pointsets in $ \mathrm{PG}(2,q^n) $ and the associated codes

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