Existence and Cardinality of k-Normal Elements in Finite Fields

Simran, Tinani,; Rosenthal, Joachim

doi:10.1007/978-3-030-68869-1_15

Cited by 3 publications

(2 citation statements)

References 18 publications

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“…In [11], the authors found a lower bound on the number of k-normal elements in F q n when they exist. We retrieve the same result using (3).…”

Section: The Number Of K-normal Elementsmentioning

confidence: 99%

Number of $k$-normal elements over a finite field

Aguirre¹,

Neumann²

2022

Preprint

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An element α ∈ F q n is a normal element over F q if the conjugates α q i , 0 ≤ i ≤ n − 1, are linearly independent over F q . Hence a normal basis for F q n over F q is of the form {α, α q , . . . , α q n−1 }, where α ∈ F q n is normal over F q .In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of knormal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of k-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).

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“…In [11], the authors found a lower bound on the number of k-normal elements in F q n when they exist. We retrieve the same result using (3).…”

Section: The Number Of K-normal Elementsmentioning

confidence: 99%

Number of $k$-normal elements over a finite field

Aguirre¹,

Neumann²

2022

Preprint

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“…Using the theory of cyclic matroids in this paper, we lay the stepping stone to answering Problem 1 stated in [20] which asks to determine the bases of α, α q , . .…”

Section: Introductionmentioning

confidence: 99%

On Cyclic Matroids and their Applications

Alfarano¹,

Khathuria²,

Simran³

2021

Preprint

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A matroid is a combinatorial structure which captures and generalizes the concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such as graphs, matrices, codes and projective geometries. In this work, we define cyclic matroids as matroids over a ground set of size n whose automorphism group contains the n-cycle. We study the properties of such matroids, with special focus on the minimum size of their bases sets. For this, we broadly employ two different approaches: the multiple basis exchange property, and an orbit-stabilizer method, developed by analyzing the action of the group Z n on the set of bases. We further present some applications of our theory to algebra and geometry, by studying the relation between cyclic matroids and cyclic projective planes, cyclic codes and k-normal elements. Finally, a general counting argument for the number of orbits of the cyclic group action is also provided.

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The average density of K-normal elements over finite fields

Reis¹

2023

Des. Codes Cryptogr.

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Existence and Cardinality of k-Normal Elements in Finite Fields

Cited by 3 publications

References 18 publications

Number of $k$-normal elements over a finite field

Number of $k$-normal elements over a finite field

On Cyclic Matroids and their Applications

The average density of K-normal elements over finite fields

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