A condition for scattered linearized polynomials involving Dickson matrices
TL;DR: In this paper, the Lunardon-Polverino binomial is shown to not be a binomial binomial if and only if x and t are linear dependent on the Dickson matrix.
Abstract: A linearized polynomial over $${{\mathbb {F}}}_{q^n}$$ is called scattered when for any $$t,x\in {{\mathbb {F}}}_{q^n}$$, the condition $$xf(t)-tf(x)=0$$ holds if and only if x and t are $${\mathbb {F}}_q$$-linearly dependent. General conditions for linearized polynomials over $${{\mathbb {F}}}_{q^n}$$ to be scattered can be deduced from the recent results in Csajbok (Scalar q-subresultants and Dickson matrices, 2018), Csajbok et al. (Finite Fields Appl 56:109–130, 2019), McGuire and Sheekey (Finite Fields Appl 57:68–91, 2019), Polverino and Zullo (On the number of roots of some linearized polynomials, 2019). 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 $$x^{q^s}+\delta x^{q^{n-s}}$$, allowing to prove that for any n and s, if $${{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1$$, 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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TL;DR: In this article, a class of scattered linearized polynomials covering infinitely many field extensions is exhibited, which leads to a new infinite family of MRD-codes in the projective line.
Abstract: A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over $${{\mathbb {F}}}_{q^6}$$
, $$q \equiv 1\pmod 4$$
described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even $$n\ge 6$$
to an $${{{\mathbb {F}}}_q}$$
-linear automorphism $$\psi (x)$$
of $${{\mathbb {F}}}_{q^n}$$
of order n. Such $$\psi (x)$$
and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line $${{\,\mathrm{{PG}}\,}}(1,q^n)$$
for $$n=8,10$$
. The polynomials described in this paper lead to a new infinite family of MRD-codes in $${{\mathbb {F}}}_q^{n\times n}$$
with minimum distance $$n-1$$
for any odd q if $$n\equiv 0\pmod 4$$
and any $$q\equiv 1\pmod 4$$
if $$n\equiv 2\pmod 4$$
.
22 citations
TL;DR: In this article, it was shown that under the action of GL ( 2, q 6 ) there are ( q 2 + q + 1 ) ( q − 2 ) / 2 ⌋ equivalence classes of maximum scattered subspaces of the form U b = { ( x, b x q + x q 4 ) : x ∈ F q 6 } in F q 2 × F q 4.
13 citations
TL;DR: This work translates the problem into the study of some algebraic curves of small degree with respect to the degree of f and implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.
11 citations
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TL;DR: The example of linear set presented is generalized to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new.
Abstract: We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered when $q$ is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters $(6,6,q;5)$.
8 citations
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TL;DR: A new family of linear maximum rank distance (MRD) codes for all parameters is constructed, which contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabdulin codes.
Abstract: In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.
211 citations
TL;DR: Some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation Ovoids of orthogonal polar spaces and finite semifields.
141 citations
TL;DR: In this paper, the dimension of a maximum scattered subspace of PG(n-1,q) with respect to a (t-1)-spread S is given, where q is a subspace intersecting every spread element in at most a point.
Abstract: A scattered subspace of PG(n-1,q) with respect to a (t-1)-spread S is a subspace intersecting every spread element in at most a point. Upper and lower bounds for the dimension of a maximum scattered space are given. In the case of a normal spread new classes of two intersection sets with respect to hyperplanes in a projective space are obtained using scattered spaces.
129 citations
TL;DR: It can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulins codes are both proper subsets of this family.
113 citations
TL;DR: Two new characterizations of the algebra Ln(Fqn) formed by all linearized polynomials over the finite field Fqn are given, and the relations between a linearization polynomial and its associated Dickson matrix are studied, generalizing a well-known criterion of Dickson on linearized permutation polynomers.
98 citations