A condition for scattered linearized polynomials involving Dickson matrices

Zanella, Corrado

doi:10.1007/s00022-019-0505-z

Cited by 24 publications

(17 citation statements)

References 22 publications

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“…Recently, different results regarding the number of roots of linearized polynomials have been presented, see [4,9,22,23,26]. In order to prove that a certain polynomial is scattered, we make use of the following result; see [4,Corollary 3.5].…”

Section: H Is Scatteredmentioning

confidence: 99%

A new family of maximum scattered linear sets in PG(1, q^6)

2020

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The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of h-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.

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Section: H Is Scatteredmentioning

confidence: 99%

A new family of maximum scattered linear sets in PG(1, q^6)

2020

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“…After a series of papers, Zanella in [39] characterized those δ ∈ F q n for which f (x) is scattered and hence both L-q t -partially and R-q t -partially scattered.…”

Section: First Examplesmentioning

confidence: 99%

“…• In Proposition 3.4 we characterize LP-polynomials which are L-q t -partially scattered or R-q t -partially scattered when n is odd. The techniques developed in [39] may be useful to extend this characterization when n is even.…”

Section: Open Problemsmentioning

confidence: 99%

Exceptional scattered polynomials

Bartoli

Zhou

2018

Journal of Algebra

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Let f be an Fq-linear function over F q n . If the Fq-subspace U = {(x q t , f (x)) : x ∈ F q n } defines a maximum scattered linear set, then we call f a scattered polynomial of index t. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function f is an exceptional scattered polynomial of index t if the subspace U associated with f defines a maximum scattered linear set in PG(1, q mn ) for infinitely many m. Our main results are the complete classifications of exceptional scattered monic polynomials of index 0 (for q > 5) and of index 1. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.

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“…This example is known as Lunardon-Polverino linear set. In [21,Theorem 3.4], Zanella proved that the condition N q n /q (δ) = 1 is necessary for L to be scattered (see also [2]). Also, a more geometric description for such linear sets has been given in [22] and an upper bound on the number of inequivalent Lunardon-Polverino linear sets has been provided in [14,Proposition 2.3], but the exact number is unknown.…”

Section: Introductionmentioning

confidence: 99%

On the automorphism groups of Lunardon-Polverino scattered linear sets

Tang¹,

Zhou²,

Zullo³

2022

Preprint

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Lunardon and Polverino introduced in 2001 a new family of maximum scattered linear sets in PG(1, q n ) to construct linear minimal Rédei blocking sets. This family has been extended first by Lavrauw, Marino, Trombetti and Polverino in 2015 and then by Sheekey in 2016 in two different contexts (semifields and rank metric codes). These linear sets are called Lunardon-Polverino linear sets and this paper aims to determine its automorphism group, to solve the equivalence issue among Lunardon-Polverino linear sets and to establish the number of inequivalent linear sets of this family. We then elaborate on this number, providing explicit bounds and determining its asymptotics.

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A condition for scattered linearized polynomials involving Dickson matrices

Cited by 24 publications

References 22 publications

A new family of maximum scattered linear sets in PG(1, q^6)

A new family of maximum scattered linear sets in PG(1, q^6)

Exceptional scattered polynomials

On the automorphism groups of Lunardon-Polverino scattered linear sets

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