Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric ...codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line
PG
(
1
,
q
n
)
. In this paper, we provide a large family of new maximum scattered linear sets over
PG
(
1
,
q
n
)
for any even
n
≥
6
and odd
q
. In particular, the relevant family contains at least
q
t
+
1
8
r
t
,
if
t
≢
2
(
mod
4
)
;
q
t
+
1
4
r
t
(
q
2
+
1
)
,
if
t
≡
2
(
mod
4
)
,
inequivalent members for given
q
=
p
r
and
n
=
2
t
>
8
, where
p
=
char
(
F
q
)
. This is a great improvement of previous results: for given
q
and
n
>
8
, the number of inequivalent maximum scattered linear sets of
PG
(
1
,
q
n
)
in all classes known so far, is smaller than
q
2
ϕ
(
n
)
/
2
, where
ϕ
denotes Euler’s totient function. Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.
Inspired by the work of Zhou (Des Codes Cryptogr 88:841–850, 2020) based on the paper of Schmidt (J Algebraic Combin 42(2):635–670, 2015), we investigate the equivalence issue of maximum
d
-codes of ...Hermitian matrices. More precisely, in the space
H
n
(
q
2
)
of Hermitian matrices over
F
q
2
we have two possible equivalences: the classical one coming from the maps that preserve the rank in
F
q
2
n
×
n
, and the one that comes from restricting to those maps preserving both the rank and the space
H
n
(
q
2
)
. We prove that when
d
<
n
and the codes considered are maximum additive
d
-codes and
(
n
-
d
)
-designs, these two equivalence relations coincide. As a consequence, we get that the idealisers of such codes are not distinguishers, unlike what usually happens for rank metric codes. Finally, we deal with the combinatorial properties of known maximum Hermitian codes and, by means of this investigation, we present a new family of maximum Hermitian 2-code, extending the construction presented by Longobardi et al. (Discrete Math 343(7):111871, 2020).
In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a
junta
(Combin ...Probab Comput 18(1–2):107–122, 2009); i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.
A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes ...of odd square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, Wilbrink’s conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).
On the List Decodability of Rank Metric Codes Trombetti, Rocco; Zullo, Ferdinando
IEEE transactions on information theory,
2020-Sept., 2020-9-00, Volume:
66, Issue:
9
Journal Article
Peer reviewed
Open access
Let <inline-formula> <tex-math notation="LaTeX">k,n,m \in {\mathbb Z} ^{+} </tex-math></inline-formula> be integers such that <inline-formula> <tex-math notation="LaTeX">k\leq n \leq m ...</tex-math></inline-formula>, let <inline-formula> <tex-math notation="LaTeX">\mathrm {G}_{n,k}\in {\mathbb F} _{q^{m}}^{n} </tex-math></inline-formula> be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>, providing <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula> is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q^{m}}^{n} </tex-math></inline-formula>. Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when <inline-formula> <tex-math notation="LaTeX">n\mid m </tex-math></inline-formula>. As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when <inline-formula> <tex-math notation="LaTeX">n\mid m </tex-math></inline-formula>, we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to <inline-formula> <tex-math notation="LaTeX">\left \lfloor{ \frac {d-1}2 }\right \rfloor +1 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to <inline-formula> <tex-math notation="LaTeX">\left \lfloor{ \frac {d-1}2 }\right \rfloor +2 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{q^{m}}^{n} </tex-math></inline-formula> of order <inline-formula> <tex-math notation="LaTeX">q^{kn} </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> dividing <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula>, such that <inline-formula> <tex-math notation="LaTeX">4n-3 </tex-math></inline-formula> is a square in <inline-formula> <tex-math notation="LaTeX">\mathbb {Z} </tex-math></inline-formula> and containing <inline-formula> <tex-math notation="LaTeX">\mathrm {G}_{n,2} </tex-math></inline-formula>, is not efficiently list decodable at some values of the radius <inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>.
Generalized twisted Gabidulin codes Lunardon, Guglielmo; Trombetti, Rocco; Zhou, Yue
Journal of combinatorial theory. Series A,
October 2018, 2018-10-00, Volume:
159
Journal Article
Peer reviewed
Open access
Let C be a set of m by n matrices over Fq such that the rank of A−B is at least d for all distinct A,B∈C. Suppose that m⩽n. If #C=qn(m−d+1), then C is a maximum rank distance (MRD for short) code. ...Until 2016, there were only two known constructions of MRD codes for arbitrary 1<d<m−1. One was found by Delsarte (1978) 8 and Gabidulin (1985) 10 independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) 16. We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) 22, and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.
In this paper, we present a new family of maximum rank-distance (MRD) codes in <inline-formula> <tex-math notation="LaTeX">\mathbb F_{q}^{2n\times 2n} </tex-math></inline-formula> of minimum distance ...<inline-formula> <tex-math notation="LaTeX">2\leq d\leq 2n </tex-math></inline-formula>. In particular, when <inline-formula> <tex-math notation="LaTeX">d=2n </tex-math></inline-formula>, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to <inline-formula> <tex-math notation="LaTeX">\mathbb F_{q^{n}} </tex-math></inline-formula>. We also prove that the MRD codes of minimum distance <inline-formula> <tex-math notation="LaTeX">2< d< 2n </tex-math></inline-formula> in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.
In this paper, we present a new family of maximum rank-distance (MRD) codes in Formula Omitted of minimum distance Formula Omitted. In particular, when Formula Omitted, we can show that the ...corresponding semifield is exactly a Hughes–Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to Formula Omitted. We also prove that the MRD codes of minimum distance Formula Omitted in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.
Generalized twisted Gabidulin codes are one of the few known families of maximum rank metric codes over finite fields. As a subset of m×n matrices, when m=n, the automorphism group of any generalized ...twisted Gabidulin code has been completely determined by the authors in 20. In this paper, we consider the same problem for m<n. Under certain conditions on their parameters, we determine their middle nuclei and right nuclei, which are important invariants with respect to the equivalence for rank metric codes. Furthermore, we also use them to derive necessary conditions on the automorphisms of generalized twisted Gabidulin codes.
On kernels and nuclei of rank metric codes Lunardon, Guglielmo; Trombetti, Rocco; Zhou, Yue
Journal of algebraic combinatorics,
09/2017, Volume:
46, Issue:
2
Journal Article
Peer reviewed
Open access
For each rank metric code
C
⊆
K
m
×
n
, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When
C
is
K
-linear, ...we also propose and investigate other two invariants called its middle nucleus and right nucleus. When
K
is a finite field
F
q
and
C
is a maximum rank distance code with minimum distance
d
<
min
{
m
,
n
}
or
gcd
(
m
,
n
)
=
1
, the kernel of the associated translation structure is proved to be
F
q
. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over
F
q
must be a finite field; its right nucleus also has to be a finite field under the condition
max
{
d
,
m
-
d
+
2
}
⩾
n
2
+
1
. Let
D
be the DHO-set associated with a bilinear dimensional dual hyperoval over
F
2
. The set
D
gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to
F
2
. Also, its middle nucleus must be a finite field containing
F
q
. Moreover, we also consider the kernel and the nuclei of
D
k
where
k
is a Knuth operation.
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