Fq-linear blocking sets in PG(2,q4)

Bonoli, Giovanna; Polverino, Olga

doi:10.2140/iig.2005.2.35

Cited by 34 publications

(38 citation statements)

References 8 publications

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“…again a contradiction. It follows that there always exists an elementλ ∈ F q 3t \ F q 3 which is not a root of g(x), and αλ satisfies Condition (7).…”

Section: Introductionmentioning

confidence: 99%

Maximum scattered linear sets and MRD-codes

et al. 2017

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The rank of a scattered F q -linear set of PG(r − 1, q n ), rn even, is at most rn/2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered F q -linear sets of rank rn/2. In this paper we prove that the bound rn/2 is sharp also in the remaining open cases.Recently Sheekey proved that scattered F q -linear sets of PG(1, q n ) of maximum rank n yield F q -linear MRD-codes with dimension 2n and minimum distance n − 1. We generalize this result and show that scattered F q -linear sets of PG(r − 1, q n ) of maximum rank rn/2 yield F q -linear MRD-codes with dimension rn and minimum distance n − 1.

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“…again a contradiction. It follows that there always exists an elementλ ∈ F q 3t \ F q 3 which is not a root of g(x), and αλ satisfies Condition (7).…”

Section: Introductionmentioning

confidence: 99%

Maximum scattered linear sets and MRD-codes

et al. 2017

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“…The above result leads, in [15], to a complete classification of all F q -linear blocking sets in PG(2, q 4 ).…”

Section: Theorem 32 ([15 Theorem 24])mentioning

confidence: 70%

“…Also, if π Λ is the quotient geometry of Σ * on Λ, then B Λ,π ,Σ is isomorphic to the F q -linear blocking set B Λ,Σ in π Λ consisting of all (n − 2)-dimensional subspaces of Σ * containing Λ and with non-empty intersection with Σ. In [15], the following has been proven.…”

Section: Linear Blocking Setsmentioning

confidence: 95%

“…All of these are Rédei type blocking sets. The F q -linear blocking sets in the plane PG (2, q 4 ) are classified in [15], where, in the table at the end of the paper, all the F q -linear blocking sets of PG(2, q 4 ), up to isomorphisms, are listed. Such a table shows that there are a lot of non-isomorphic families of F q -linear blocking sets in PG(2, q 4 ), both of Rédei and of non-Rédei type.…”

Section: Linear Blocking Setsmentioning

confidence: 99%

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Linear sets in finite projective spaces

Polverino¹

2010

Discrete Mathematics

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a b s t r a c tIn this paper linear sets of finite projective spaces are studied and the ''dual'' of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides ''old'' results, new ones are proven and some open questions are discussed.

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“…For linear sets of rank n in PG(1, q n ), Theorem 1.2 was shown in [3,Lemma 2.2]. In Section 2, the connection between linear sets in PG(1, q n ) and the direction problem is described.…”

Section: Introductionmentioning

confidence: 99%

The minimum size of a linear set

Beule

Voorde

2019

Journal of Combinatorial Theory, Series A

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In this paper, we first determine the minimum possible size of an F q -linear set of rank k in PG(1, q n ). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F qlinear set of rank k in PG(2, q n ). In the case k = n, this confirms a conjecture by Sziklai in [9].

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Fq-linear blocking sets in PG(2,q4)

Cited by 34 publications

References 8 publications

Maximum scattered linear sets and MRD-codes

Maximum scattered linear sets and MRD-codes

Linear sets in finite projective spaces

The minimum size of a linear set

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