MRD codes with maximum idealizers

Csajbók, Bence; Marino, Giuseppe; Polverino, Olga; Zhou, Yue

doi:10.1016/j.disc.2020.111985

Cited by 23 publications

(11 citation statements)

References 46 publications

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“…Notice that this assumption on t is taken in order to ease the computations, even though the codes C h,t,σ are MRD also for t = 3, 4. When t = 3 the inequivalence with the other known F q n -linear MRD codes such as Gabidulin codes, twisted Gabidulin codes and those in [6,7,10,28,47] has been proved in [3,Section 4]. However when t ∈ {3, 4} the computations of this section become more complicated, since some of the arguments that we are going to use do not work.…”

Section: Study Of the Equivalence Of The New Familymentioning

confidence: 96%

“…Further examples of F q n -linear MRD codes can be found in [3,6,7,10,28,47] which exist only for n ∈ {6, 7, 8} and in [23,24] which exist for every n even.…”

Section: Gabidulin and Twisted Gabidulin Codesmentioning

confidence: 99%

“…Another large family of MRD codes was later given by Trombetti and Zhou in [45]. Except from that, almost all the other known MRD construction have all very specific restriction on some of the parameters: for instance, the MRD codes [3,6,7,10,28,47] exist only for n ∈ {6, 7, 8}.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extending two families of maximum rank distance codes

Neri¹,

Santonastaso²,

Zullo³

2021

Preprint

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In this paper we provide a large family of rank-metric codes, which contains properly the codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are F q 2t -linear of dimension 2 in the space of linearized polynomials over F q 2t , where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t ≥ 5, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.

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Section: Study Of the Equivalence Of The New Familymentioning

confidence: 96%

“…Further examples of F q n -linear MRD codes can be found in [3,6,7,10,28,47] which exist only for n ∈ {6, 7, 8} and in [23,24] which exist for every n even.…”

Section: Gabidulin and Twisted Gabidulin Codesmentioning

confidence: 99%

See 1 more Smart Citation

Extending two families of maximum rank distance codes

Neri¹,

Santonastaso²,

Zullo³

2021

Preprint

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“…In order to do this, we explicit the coefficients c d−1,0 and c d−1,d−1 . Indeed, the coefficient c k d−1,0 can be obtained by applying (8) with t = 0 and j…”

Section: Some Necessary Conditionsmentioning

confidence: 99%

“…Then we use 3. of Theorem 1.3 to introduce a new class of σ-linearized trinomials with maximum kernel when q is even and we explicit such equations when d = 3 and d = 4. Finally we apply these results to investigate rank metric codes introduced in [8], to construct examples of quasi subfield polynomials and of cyclic subspace codes. These constructions of subspace codes yield new solutions to the conjecture posed in [35].…”

Section: Introductionmentioning

confidence: 99%

Linearized trinomials with maximum kernel

Santonastaso¹,

Zullo²

2020

Preprint

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Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let q be a prime power, n be a positive integer and σ be a generator of Gal(F q n : F q ). In this paper we provide closed formulas for the coefficients of a σ-trinomial f over F q n which ensure that the dimension of the kernel of f equals its σ-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having σ-degree 3 and 4. Our techniques rely on the tools developed in [22]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8], to construct quasi subfield polynomials and cyclic subspace codes, obtaining new solutions to the conjecture posed in [35].

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Generalised evasive subspaces

Gruica,

Ravagnani,

Sheekey

et al. 2024

J of Combinatorial Designs

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We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank‐metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank‐metric codes.

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MRD codes with maximum idealizers

Cited by 23 publications

References 46 publications

Extending two families of maximum rank distance codes

Extending two families of maximum rank distance codes

Linearized trinomials with maximum kernel

Generalised evasive subspaces

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