Linear sets on the projective line with complementary weights

Napolitano, Vito; Polverino, Olga; Santonastaso, Paolo; Zullo, Ferdinando

doi:10.48550/arxiv.2107.10641

Cited by 4 publications

(11 citation statements)

References 25 publications

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“…where ξ 2 = aξ + b and η ρ = Aξ + B with a, b, A, B ∈ F q t . Using the above result, similarly to what has been done in [24,Theorem 4.14], we obtain the following result.…”

Section: Examples Of Maximum Multi-sidon Spacessupporting

confidence: 75%

“…Now, we will study the equivalence issue among this family of multi-Sidon spaces. To this aim, we start by recalling the following lemma proved in [24]. Lemma 5.2.…”

Section: Examples Of Maximum Multi-sidon Spacesmentioning

confidence: 99%

“…Linear sets of this form in PG(1, q n ) have been already studied in [24], where the authors study linear sets on the projective line containing two points whose sum of their weights equals the rank of the linear set.…”

Section: Linear Sets With Only R Points Of Weight Greater Than Onementioning

confidence: 99%

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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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“…where ξ 2 = aξ + b and η ρ = Aξ + B with a, b, A, B ∈ F q t . Using the above result, similarly to what has been done in [24,Theorem 4.14], we obtain the following result.…”

Section: Examples Of Maximum Multi-sidon Spacessupporting

confidence: 75%

“…Now, we will study the equivalence issue among this family of multi-Sidon spaces. To this aim, we start by recalling the following lemma proved in [24]. Lemma 5.2.…”

Section: Examples Of Maximum Multi-sidon Spacesmentioning

confidence: 99%

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Multi-Sidon spaces over finite fields

Zullo¹

2021

Preprint

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“…The examples of linear sets in Theorem 2.5 admits two points with complementary weights, that is the sum of the weights of such points equals the rank of the linear set. These type of linear sets have been recently investigated in [21]. We recall a useful result from [21], regarding the number of points of a linear set with two points with complementary weights.…”

Section: Linear Setsmentioning

confidence: 99%

“…These type of linear sets have been recently investigated in [21]. We recall a useful result from [21], regarding the number of points of a linear set with two points with complementary weights.…”

Section: Linear Setsmentioning

confidence: 99%

Classifications and constructions of minimum size linear sets

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2022

Preprint

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This paper aims to study linear sets of minimum size in the projective line, that is F q -linear sets of rank k in PG(1, q n ) admitting one point of weight one and having size q k−1 + 1. Examples of these linear sets have been found by Lunardon and the second author ( 2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for k ≤ 5. In this paper we provide classification results for those L U admitting two points with complementary weights. We construct new examples and also study the related ΓL(2, q n )-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.

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The geometric field of linearity of linear sets

Jena

Voorde

2022

Des. Codes Cryptogr.

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If an $${\mathbb {F}}_q$$ F q -linear set $$L_U$$ L U in a projective space is defined by a vector subspace U which is linear over a proper superfield of $${\mathbb {F}}_{q}$$ F q , then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank h in $$\mathrm {PG}(1,q^h)$$ PG ( 1 , q h ) but for linear sets of rank $$k<h$$ k < h the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the geometric field of linearity of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.

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Linear sets on the projective line with complementary weights

Cited by 4 publications

References 25 publications

Multi-Sidon spaces over finite fields

Multi-Sidon spaces over finite fields

Classifications and constructions of minimum size linear sets

The geometric field of linearity of linear sets

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