This paper is devoted to introduce the structure of the p-ary linear codes C(n,q) of points and lines of PG(n,q),q=p^h prime. When p=3, the linear code C(2,27) is given with its generator matrix and ...also, some of weight distributions are calculated.
A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the
q
-polynomial over
F
q
6
,
q
≡
1
(
mod
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
≥
6
to an
F
q
-linear automorphism
ψ
(
x
)
of
F
q
n
of order
n
. Such
ψ
(
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
PG
(
1
,
q
n
)
for
n
=
8
,
10
. The polynomials described in this paper lead to a new infinite family of MRD-codes in
F
q
n
×
n
with minimum distance
n
-
1
for any odd
q
if
n
≡
0
(
mod
4
)
and any
q
≡
1
(
mod
4
)
if
n
≡
2
(
mod
4
)
.
On the equivalence of linear sets Csajbók, Bence; Zanella, Corrado
Designs, codes, and cryptography,
11/2016, Letnik:
81, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Let
L
be a linear set of pseudoregulus type in a line
ℓ
in
Σ
∗
=
PG
(
t
-
1
,
q
t
)
,
t
=
5
or
t
>
6
. We provide examples of
q
-order canonical subgeometries
Σ
1
,
Σ
2
⊂
Σ
∗
such that there is a
(
t
...-
3
)
-subspace
Γ
⊂
Σ
∗
\
(
Σ
1
∪
Σ
2
∪
ℓ
)
with the property that for
i
=
1
,
2
,
L
is the projection of
Σ
i
from center
Γ
and there exists no collineation
ϕ
of
Σ
∗
such that
Γ
ϕ
=
Γ
and
Σ
1
ϕ
=
Σ
2
. Condition (ii) given in Theorem
3
in Lavrauw and Van de Voorde (Des Codes Cryptogr 56:89–104,
2010
) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.
A linearized polynomial f(x)∈Fqnx is called scattered if for any y,z∈Fqn, the condition zf(y)−yf(z)=0 implies that y and z are Fq-linearly dependent. In this paper two generalizations of the notion ...of a scattered linearized polynomial are provided and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-qt-partially scattered and R-qt-partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-qt- and R-qt-partially scattered. Also, connections between partially scattered polynomials, linear sets and rank metric codes are exhibited.
Consider the n-dimensional projective space PG(n,q) over a finite field with q elements. A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition of the set ...of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG(n,q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q2. In the present paper we construct parallelisms of PG(3,7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG(3,7). The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one.
An
m-cover
of lines of a finite projective space
PG
(
r
,
q
)
(of a finite polar space
P
) is a set of lines
L
of
PG
(
r
,
q
)
(of
P
) such that every point of
PG
(
r
,
q
)
(of
P
) contains
m
lines ...of
L
, for some
m
. Embed
PG
(
r
,
q
)
in
PG
(
r
,
q
2
)
. Let
L
¯
denote the set of points of
PG
(
r
,
q
2
)
lying on the extended lines of
L
. An
m
-cover
L
of
PG
(
r
,
q
)
is an
(
r
-
2
)
-dual
m
-cover if there are two possibilities for the number of lines of
L
contained in an
(
r
-
2
)
-space of
PG
(
r
,
q
)
. Basing on this notion, we characterize
m
-covers
L
of
PG
(
r
,
q
)
such that
L
¯
is a two-character set of
PG
(
r
,
q
2
)
. In particular, we show that if
L
is invariant under a Singer cyclic group of
PG
(
r
,
q
)
then it is an
(
r
-
2
)
-dual
m
-cover. Assuming that the lines of
L
are lines of a symplectic polar space
W
(
r
,
q
)
(of an orthogonal polar space
Q
(
r
,
q
)
of parabolic type), similarly to the projective case we introduce the notion of an
(
r
-
2
)
-dual
m
-cover of symplectic type (of parabolic type). We prove that an
m
-cover
L
of
W
(
r
,
q
)
(of
Q
(
r
,
q
)
) has this dual property if and only if
L
¯
is a tight set of an Hermitian variety
H
(
r
,
q
2
)
or of
W
(
r
,
q
2
)
(of
H
(
r
,
q
2
)
or of
Q
(
r
,
q
2
)
). We also provide some interesting examples of
(
4
n
-
3
)
-dual
m
-covers of symplectic type of
W
(
4
n
-
1
,
q
)
.
A
k
-arc in the projective space
PG
(
n
,
q
)
is a set of
k
projective points such that no subcollection of
n
+
1
points is contained in a hyperplane. In this paper, we construct new 60-arcs and ...110-arcs in
PG
(
4
,
q
)
that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set
P
of projective points in the projective space of dimension
n
over an algebraic number field
Q
(
ξ
)
, determines a complete list of primes
p
for which the reduction modulo
p
of
P
to the projective space
PG
(
n
,
p
h
)
may fail to be a
k
-arc. Using these methods, we prove that there are infinitely many primes
p
such that
PG
(
4
,
p
)
contains a
PSL
(
2
,
11
)
-invariant 110-arc, where
PSL
(
2
,
11
)
is given in one of its natural irreducible representations as a subgroup of
PGL
(
5
,
p
)
. Similarly, we show that there exist
PSL
(
2
,
11
)
-invariant 110-arcs in
PG
(
4
,
p
2
)
and
PSL
(
2
,
11
)
-invariant 60-arcs in
PG
(
4
,
p
)
for infinitely many primes
p
.
In this note, the existence of 66,10,363 and 55,15,243 linear codes is proven. These ternary codes are constructed from orbits of a projectivity group of PG(k−1,F3), for k=10,15. These new codes and ...their punctured subcodes improve the best known lower bounds on the largest possible minimum distance.
Let
L
=
F
q
n
be a finite field and let
F
=
F
q
be a subfield of
L
. Consider
L
as a vector space over
F
and the associated projective space that is isomorphic to PG(
n
− 1,
q
). The properties of ...the projective mapping induced by
x
↦
x
-
1
have been studied in Csajbók (Finite Fields Appl. 19:55–66,
2013
), Faina et al. (Eur. J. Comb. 23:31–35,
2002
), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266–275,
1983
), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211–228
1985
, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam,
1995
). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer
k
, if
q
≥ 2
k
− 1, then there are partitions of PG(2
k
− 1,
q
) in normal rational curves of degree 2
k
− 1. For smaller
q
the same construction gives partitions in (
q
+ 1)-tuples of independent points.
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