On linear sets of minimum size

Jena, Dibyayoti; Voorde, Geertrui Van de

doi:10.1016/j.disc.2020.112230

Cited by 10 publications

(19 citation statements)

References 20 publications

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“…In this latter case, we prove that they all correspond to the construction found in [11], whereas in the former case we provide more examples. Then we introduce a general procedure to construct examples of linear sets in PG(1, q n ) starting from a linear set L U ′ in PG(1, q t ), with t | n, whose weight spectrum is completely determined by the weight distribution of L U ′ .…”

Section: Introductionsupporting

confidence: 57%

“…As a corollary we can refine the classification of minimum size linear sets of rank 5 given in [11,Proposition 2.12]. Corollary 3.9.…”

Section: Linear Sets With Complementary Weights Of Type (K − 2 2)mentioning

confidence: 77%

“…+ x q n−1 , which has size q n−1 + 1, q n−1 points of weight one and one point of weight n − 1. Other examples of such linear sets with maximum rank n have been found in [19], recently generalized by Jena and Van de Voorde [11]. Minimum size linear sets in PG(1, q n ) of rank at most 5 have been classified and they all correspond to the linear sets described above.…”

Section: Introductionmentioning

confidence: 99%

“…Then we introduce a general procedure to construct examples of linear sets in PG(1, q n ) starting from a linear set L U ′ in PG(1, q t ), with t | n, whose weight spectrum is completely determined by the weight distribution of L U ′ . In particular, using the family of minimum size linear sets of [11], we are able to get more examples of minimum size linear sets different from those found in [11] and also with a different weight spectrum. We then study the ΓL(2, q n )-equivalence among the subspaces defining the linear set of [11] and those found in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, using the family of minimum size linear sets of [11], we are able to get more examples of minimum size linear sets different from those found in [11] and also with a different weight spectrum. We then study the ΓL(2, q n )-equivalence among the subspaces defining the linear set of [11] and those found in this paper. These results answer to the open problem posed in [11, (D) Section 2.5].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionsupporting

confidence: 57%

“…As a corollary we can refine the classification of minimum size linear sets of rank 5 given in [11,Proposition 2.12]. Corollary 3.9.…”

Section: Linear Sets With Complementary Weights Of Type (K − 2 2)mentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classifications and constructions of minimum size linear sets

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2022

Preprint

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The geometric field of linearity of linear sets

Jena

Voorde

2022

Des. Codes Cryptogr.

Self Cite

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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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Constructions and equivalence of Sidon spaces

Castello,

Polverino,

Santonastaso

et al. 2023

J Algebr Comb

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Sidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017), in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of $$\mathbb {F}_{q^n}$$ F q n , the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of $$\mathbb {F}_{q^n}$$ F q n and finally the ones obtained as the kernel of subspace polynomials. In this paper, we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples.

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On linear sets of minimum size

Cited by 10 publications

References 20 publications

Classifications and constructions of minimum size linear sets

Classifications and constructions of minimum size linear sets

The geometric field of linearity of linear sets

Constructions and equivalence of Sidon spaces

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