Scattered subspaces and related codes Zini, Giovanni; Zullo, Ferdinando
Designs, codes, and cryptography,
08/2021, Volume:
89, Issue:
8
Journal Article
Peer reviewed
After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum
h
-scattered
F
q
-subspaces of
V
(
r
,
q
n
)
and maximum rank distance (MRD) codes has been ...established in the extremal cases
h
=
1
and
h
=
r
-
1
. In this paper, we propose a connection for any
h
∈
{
1
,
…
,
r
-
1
}
, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an
h
-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum
h
-scattered subspaces.
In this paper we consider a family
F
of 2
n
-dimensional
F
q
-linear rank metric codes in
F
q
n
×
n
arising from polynomials of the form
x
q
s
+
δ
x
q
n
2
+
s
∈
F
q
n
x
. The family
F
was ...introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that
F
contains MRD codes for
n
=
8
, and other subsequent partial results have been provided in the literature towards the classification of MRD codes in
F
for any
n
. In particular, the classification has been reached when
n
is smaller than 8, and also for
n
greater than 8 provided that
s
is small enough with respect to
n
. In this paper we deal with the open case
n
=
8
, providing a classification for any large enough odd prime power
q
. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional
F
q
-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in
F
are not equivalent to any other MRD codes known so far.
In this paper, we deal with the problem of classifying the genera of quotient curves
H
q
/
G
, where
H
q
is the
F
q
2
-maximal Hermitian curve and
G
is an automorphism group of
H
q
. The groups
G
...considered in the literature fix either a point or a triangle in the plane
PG
(
2
,
q
6
)
. In this paper, we give a complete list of genera of quotients
H
q
/
G
, when
G
≤
Aut
(
H
q
)
≅
PGU
(
3
,
q
)
does not leave invariant any point or triangle in the plane. Also, the classification of subgroups
G
of
PGU
(
3
,
q
)
satisfying this property is given up to isomorphism.
In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided. Our method is based on the CSS construction as well as on some peculiar properties of the ...underlying algebraic curves, named Swiss curves. Several classes of well-known algebraic curves with many rational points turn out to be Swiss curves. Examples are given by Castle curves, GK curves, generalized GK curves and the Abdón–Bezerra–Quoos maximal curves. Applications of our method to these curves are provided. Our construction extends a previous one due to Hernando, McGuire, Monserrat, and Moyano-Fernández.
We investigate the genera of quotient curves ℋ
q
∕G of the
-maximal Hermitian curve ℋ
q
, where G is contained in the maximal subgroup
fixing a pole-polar pair (P,ℓ) with respect to the unitary ...polarity associated with ℋ
q
. To this aim, a geometric and group-theoretical description of ℳ
q
is given. The genera of some other quotients ℋ
q
∕G with G≰ℳ
q
are also computed. In this way we obtain new values in the spectrum of genera of
-maximal curves. The complete list of genera g>1 of quotients of ℋ
q
is given for q≤29, as well as the genera g of quotients of ℋ
q
with
for any q. As a direct application, we exhibit examples of
-maximal curves which are not Galois covered by ℋ
q
when q is not a cube. Finally, a plane model for ℋ
q
∕G is obtained when G is cyclic of order p⋅d, with d a divisor of q+1.
The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with ...possibly different maximum fields of linearity. We also consider the intersection between a certain linear set of maximum rank and any other linear set of the same rank. The strategy relies on the study of certain algebraic curves whose rational points describe the intersection of the two linear sets. Among other geometric and algebraic tools, function field theory and the Hasse–Weil bound play a crucial role. As an application, we give asymptotic results on semifields of BEL-rank two.
Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by ...the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q^{n}} </tex-math></inline-formula>-linear maximum rank distance codes, introducing the notion of exceptional linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.
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