The weight distributions of linear sets in PG(1,q^5)

Boeck, Maarten De; Voorde, Geertrui Van de

doi:10.48550/arxiv.2006.04961

Cited by 3 publications

(5 citation statements)

References 6 publications

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“…In [24] the authors study the weight distribution of linear sets in PG(1, q 5 ). In particular, they prove that linear sets of rank 5 in PG(1, q 5 ) containing one point of weight 3, one point of weight 2 and all the others of weight one do not exist.…”

Section: Linear Sets With Exactly Two Points Of Weight Greater Than Onementioning

confidence: 99%

Linear sets on the projective line with complementary weights

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2021

Preprint

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Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021).

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Section: Linear Sets With Exactly Two Points Of Weight Greater Than Onementioning

confidence: 99%

Linear sets on the projective line with complementary weights

Napolitano¹,

Polverino²,

Santonastaso³

et al. 2021

Preprint

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“…The following lemma provides a way to find such lines. The first part of this lemma was already proven for linear sets in PG(1, q 5 ) in [8].…”

Section: Constructing 5-linesmentioning

confidence: 82%

“…A point of lying on the extension of a -line is a point of rank 2. Definition 2.2 (see also [8]) Let Q be a point of of rank 2 lying on the extension of a -line containing two points P 1 = v 1 and P 2 = v 2 , where P 1 , P 2 ∈ , then Q = v 1 − αv 2 for some α ∈ F q h \ F q . We say that Q has type S α , where…”

Section: -Lines and Their Typementioning

confidence: 99%

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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“…By direct computation using w 2 = w + 1, all the q 2 distinct solutions of (15) are in F q 5 and every b satisfying (15) satisfies also (14). Consider System (13). Similar arguments show that a = d = 0, a contradiction.…”

Section: Known Examples Of 1-fat Polynomialsmentioning

confidence: 91%

“…2, L U,v Fq is PΓL-equivalent to one of L B,1 , L B,21 , L B,22 , L C,11 , L C,12 , L C,13 , L C,14 and L C,15 as defined in Theorem 5.2. By[27, Theorem 5], the F q -linear blocking sets with more than one Rédei line are exactly the ones which are PΓL-equivalent to L B,1 .…”

mentioning

confidence: 97%

$r$-fat linearized polynomials over finite fields

Bartoli¹,

Micheli²,

Zini³

et al. 2020

Preprint

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In this paper we prove that the property of being scattered for a F q -linearized polynomial of small q-degree over a finite field F q n is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate r-fat polynomials, a natural generalization of scattered polynomials. An r-fat F q -linearized polynomial defines a linear set of rank n in the projective line of order q n with r points of weight larger than one. When r equals 1, the corresponding linear sets are called clubs, and they are related with a number of remarkable mathematical objects like KM-arcs, group divisible designs and rank metric codes. Using techniques on algebraic curves and global function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r > 0. In the case n ≤ 4, we completely determine the spectrum of values of r for which an r-fat polynomial exists. In the case n = 5, we provide a new family of 1-fat polynomials. Furthermore, we determine the values of r for which the so-called LP-polynomials are r-fat.

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The weight distributions of linear sets in PG(1,q^5)

Abstract: In this paper, we determine all possible weight distributions of F q -linear sets properly contained in PG(1, q 5 ). In particular, we show that there exist no 2-clubs of rank 5, and more generally, that there are no F q -linear sets of rank 5 containing few points of weight 2.

Cited by 3 publications

References 6 publications

Linear sets on the projective line with complementary weights

Linear sets on the projective line with complementary weights

The geometric field of linearity of linear sets

$r$-fat linearized polynomials over finite fields

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