Daniele Bartoli
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View article: Long QMDS additive code
Long QMDS additive code Open
We investigate additive codes, defined as $\mathbb{F}_q$-linear subspaces $C \subseteq \mathbb{F}_{q^h}^n$ of length $n$ and dimension $r$ over $\mathbb{F}_q$. An additive code is said to be of type $[n, r/h, d]_q^h$, where $d$ denotes the…
Complete $$(k,q+1)$$-arcs in $$\textrm{PG}(2,\mathbb {F}_{q^6})$$ from the Hermitian curve Open
We prove that, if q is large enough, the set of the $$\mathbb {F}_{q^6}$$ -rational points of the Hermitian curve is a complete $$(q^6+q^5-q^4+1,q+1)$$ -arc in $$\textrm{PG}(2,\mathbb {F}_{q^6})$$ , addressing an open case from a recent pa…
View article: Linear rank-metric intersecting codes
Linear rank-metric intersecting codes Open
In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said …
On APN functions in odd characteristic, the disproof of a conjecture and related problems Open
In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality $q$ is larger than~$9587$, then those functions will never be APN. Moreover,…
View article: Towards the classification of scattered binomials
Towards the classification of scattered binomials Open
Let \( q \) be a prime power and \( n \) an integer. An \( \mathbb{F}_q \)-linearized polynomial \( f \) is said to be scattered if it satisfies the condition that for all \( x, y \in \mathbb{F}_q^n \setminus \{ 0 \} \), whenever \( \frac{…
Ovoids of $Q^+(7,q)$ of low-degree Open
Ovoids of the hyperbolic quadric $Q^+(7,q)$ of $\mathrm{PG}(7,q)$ have been extensively studied over the past 40 years, partly due to their connections with other combinatorial objects. It is well known that the points of an ovoid of $Q^+(…
Exceptional scattered sequences Open
The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of -linear MRD codes. The first infinite family in the first nontrivial case is also pr…
A proof of a conjecture on trivariate permutations Open
In this note we show (for a large enough dimension of the underlying field) a conjecture of [C. Beierle, C. Carlet, G. Leander, L. Perrin, {\em A further study of quadratic APN permutations in dimension nine}, Finite Fields Appl. 81 (2022)…
A proof of a conjecture on permutation trinomials Open
In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +αX^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$, from …
View article: New scattered subspaces in higher dimensions
New scattered subspaces in higher dimensions Open
Over the past few decades, there has been extensive research on scattered subspaces, partly because of their link to MRD codes. These subspaces can be characterized using linearized polynomials over finite fields. Within this context, scat…
A new infinite family of maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(m(h+1),q^n)$ and associated MRD codes Open
The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces pla…
Ovoids of Q(6, q) of low degree Open
Ovoids of the parabolic quadric Q (6, q ) of $$\textrm{PG}(6,q)$$ have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q (6, q ), the Thas-Kantor ovoid…
On $3$-dimensional MRD codes of type $\langle x^{q^t},x+δx^{q^{2t}},G(x) \rangle$ Open
In this work we present results on the classification of $\mathbb{F}_{q^n}$-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD…
View article: A new family of $2$-scattered subspaces and related MRD codes
A new family of $2$-scattered subspaces and related MRD codes Open
Scattered subspaces and $h$-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist, relativ…
New scattered subspaces in higher dimensions Open
Over the past few decades, there has been extensive research on scattered subspaces, partly because of their link to MRD codes. These subspaces can be characterized using linearized polynomials over finite fields. Within this context, scat…
On the classification of low degree ovoids of $Q^+(5,q)$ Open
Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a t…
View article: Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic
Scattered trinomials of $\mathbb{F}_{q^6}[X]$ in even characteristic Open
In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [B. Csajbók, G. Marino and F. Zullo: New maximum scattered linear sets of the p…
View article: Saturating linear sets of minimal rank
Saturating linear sets of minimal rank Open
Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…
On Permutation Trinomials of the type $X^{q^2-q+1}+AX^{q^2}+BX$ over $\mathbb{F}_{q^3}$ Open
Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.
New scattered sequences of order 3 Open
Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infinite family of scattered sequences of order three is obtained. Equivalence…
Complete $(q+1)$-arcs in $\mathrm{PG}(2,\mathbb{F}_{q^6})$ from the Hermitian curve Open
We prove that, if $q$ is large enough, the set of the $\mathbb{F}_{q^6}$-rational points of the Hermitian curve is a complete $(q+1)$-arc in $\mathrm{PG}(2,\mathbb{F}_{q^6})$, addressing an open case from a recent paper by Korchmáros, Szőn…
On the exceptionality of rational APN functions Open
We investigate APN functions which can be represented as rational functions and we provide non-existence results exploiting the connection between these functions and specific algebraic varieties over finite fields. This approach allows to…
Investigating rational perfect nonlinear functions Open
Perfect nonlinear (PN) functions over a finite field, whose study is also motivated by practical applications to Cryptography, have been the subject of several recent papers where the main problems, such as effective constructions and non-…
Linear Maximum Rank Distance Codes of Exceptional Type Open
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 par…
Small Strong Blocking Sets by Concatenation Open
Strong blocking sets and their counterparts, minimal codes, attracted lots of\nattention in the last years. Combining the concatenating construction of codes\nwith a geometric insight into the minimality condition, we explicitly provide\ni…
Preface Open
Ever since their first mention, error correcting codes have played an important role in modern communication.Over the last few decades, they have gained in importance due to the ever-increasing amount of data that we store and communicate …
Exceptional scattered sequences Open
The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of $\mathbb{F}_{q^n}$-linear MRD codes. The first infinite family in the first nontrivi…