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Zanella, Corrado; Zullo, Ferdinando

doi:10.1016/j.disc.2019.111800

Cited by 30 publications

(24 citation statements)

References 29 publications

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“…Note that in Case 1, this example coincides with the one introduced in [27], where it has been proved that f h is scattered for q ≡ 1 (mod 4) and q ≤ 29. In Corollary 3.11 we will prove that the linear set L h associated with f h (x) is new, apart from the case of q a power of 5 and h ∈ F q .…”

Section: Introductionsupporting

confidence: 71%

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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: Introductionsupporting

confidence: 71%

“…In Corollary 3.11 we will prove that the linear set L h associated with f h (x) is new, apart from the case of q a power of 5 and h ∈ F q . This solves an open problem posed in [27].…”

Section: Introductionmentioning

confidence: 87%

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

2020

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“…The theorem above has been proved in the particular cases n = 4 in [10], s = 1 in [2], both n and q odd in [11], and odd n in [22].…”

Section: A Condition For Scattered Linearized Polynomialsmentioning

confidence: 91%

“…In particular cases, the condition N q n /q (δ) = 1 has been proved to be necessary for f (x) to be scattered [2,10,11,22]. In section 3 it will proved that actually it is necessary for any n and s. Further examples of scattered q-polynomials are given in [6,5,14,22]. All of them are with respect to n = 6 or n = 8.…”

Section: Introductionmentioning

confidence: 99%

A condition for scattered linearized polynomials involving Dickson matrices

Zanella

2019

J. Geom.

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A linearized polynomial over F q n is called scattered when for any t, x ∈ F q n , the condition xf (t) − tf (x) = 0 holds if and only if x and t are F q -linearly dependent. General conditions for linearized polynomials over F q n to be scattered can be deduced from the recent results in [4,7,15,19]. Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon-Polverino binomial x q s + δx q n−s , allowing to prove that for any n and s, if N q n /q (δ) = 1, then the binomial is not scattered. Also, a necessary and sufficient condition for x q s +bx q 2s to be scattered is shown which is stated in terms of a special plane algebraic curve.

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“…• Non-linear MRD codes by Cossidente, the second author and Pavese [5] which were later generalized by Durante and Siciliano [17]. • Linear MRD codes associated with maximum scattered linear sets of PG(1, q 6 ) and PG(1, q 8 ) presented recently in [1,7,9,38,54].…”

Section: Introductionmentioning

confidence: 99%

MRD codes with maximum idealizers

Csajbók

Marino

Polverino

et al. 2020

Discrete Mathematics

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Left and right idealizers are important invariants of linear rankdistance codes. In the case of maximum rank-distance (MRD for short) codes in F n×n q the idealizers have been proved to be isomorphic to finite fields of size at most q n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in F n×n q for n ≤ 9 with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rankdistance codes providing MRD ones with maximum idealizers for n = 7, q odd and for n = 8, q ≡ 1 (mod 3). These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for n ≥ 9.

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Vertex properties of maximum scattered linear sets of PG(1,qn)

Cited by 30 publications

References 29 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)

A condition for scattered linearized polynomials involving Dickson matrices

MRD codes with maximum idealizers

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