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Daniele Bartoli

Publications and source records attributed to Daniele Bartoli.

At least 37 records · Page 2Linked to original sources
arXiv2024

On $3$-dimensional MRD codes of type $\langle x^{q^t},x+δx^{q^{2t}},G(x) \rangle$

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 codes $\mathcal{C}=\langle x^{q^t}, F(x), G(x) \rangle \subseteq \mathcal{L}_{n,q}$ of exceptional type, i.e. such that $\mathcal{C}$ is MRD over infinite many extensions of the field $\mathbb{F}_{q^n}$. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.

cs.IT↗
arXiv2024

A new family of $2$-scattered subspaces and related MRD codes

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, relatively few are known about $h$-scattered subspaces with $h\geq2$. In this paper, we establish the existence of maximum $2$-scattered $\F_q$-subspaces in $V(r,q^6)$ whenever $r\geq 3$, $r\ne 5$, and $q$ is an odd power of $2$. Additionally, we explore the corresponding MRD codes.

math.CO↗
arXiv2023

On the classification of low degree ovoids of $Q^+(5,q)$

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 three dimensional elliptic quadric (corresponding to the regular spread of $\mathrm{PG}(3,q)$) many other classes of examples are known. First of all the other examples (beside the elliptic quadric) of ovoids of $Q(4,q)$ give also examples of ovoids of $Q^+(5,q)$. Another important class of ovoids of $Q^+(5,q)$ is given by the ones associated to a flock of a three dimensional quadratic cone. To every ovoid of $Q^+(5,q)$ two bivariate polynomials $f_1(x,y)$ and $f_2(x,y)$ can be associated. In this paper, we classify ovoids of $Q^+(5,q)$ such that $f_1(x,y)=y+g(x)$ and $\max\{deg(f_1),deg(f_2)\}<(\frac{1}{6.3}q)^{\frac{3}{13}}-1$, that is $f_1(x,y)$ and $f_2(x,y)$ have "low degree" compared with $q$.

math.CO↗
arXiv2023

Saturating linear sets of minimal rank

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 paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

math.CO↗
arXiv2023

New scattered sequences of order 3

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 issues are also considered.

math.CO↗
arXiv2023

Complete $(q+1)$-arcs in $\mathrm{PG}(2,\mathbb{F}_{q^6})$ from the Hermitian curve

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, Szőnyi and Nagy. An algebraic approach based on the investigation of some algebraic varieties attached to the arc is used.

math.CO↗
arXiv2023

Small Strong Blocking Sets by Concatenation

Strong blocking sets and their counterparts, minimal codes, attracted lots of attention in the last years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.

math.CO↗
arXiv2022

Exceptional scattered sequences

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 nontrivial case is also provided and equivalence issues are considered. As a byproduct, a new infinite family of MRD codes is obtained.

math.CO↗
arXiv2022

Minimal codewords in Norm-Trace codes

In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.

math.AG↗
arXiv2022

New MRD codes from linear cutting blocking sets

Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we provide the first construction of a family of $\mathbb{F}_{q^m}$-linear MRD codes of length $2m$ that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.

math.CO↗
arXiv2022

Differential biases, $c$-differential uniformity, and their relation to differential attacks

Differential cryptanalysis famously uses statistical biases in the propagation of differences in a block cipher to attack the cipher. In this paper, we investigate the existence of more general statistical biases in the differences. To this end, we discuss the $c$-differential uniformity of S-boxes, which is a concept that was recently introduced in Ellingsen et. al. to measure certain statistical biases that could potentially be used in attacks similar to differential attacks. Firstly, we prove that a large class of potential candidates for S-boxes necessarily has large $c$-differential uniformity for all but at most $B$ choices of $c$, where $B$ is a constant independent of the size of the finite field $q$. This result implies that for a large class of functions, certain statistical differential biases are inevitable. In a second part, we discuss the practical possibility of designing a differential attack based on weaknesses of S-boxes related to their $c$-differential uniformity.

cs.CR↗
arXiv2022

Towards the classification of exceptional scattered polynomials

Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered polynomials, i.e. polynomials which are scattered over infinite field extensions, are limited to the cases where their index $\ell$ is small, or a prime number larger than the $q$-degree $k$ of the polynomial, or an integer smaller than the $k$ in the case where $k$ is a prime. In this paper we completely classify exceptional scattered polynomials when the maximum between $\ell$ and $k$ is odd, and give partial results when it is even, extending a result of Ferraguti and Micheli in 2021.

math.NT↗
arXiv2022

On the classification of low-degree ovoids of Q(4,q)

Ovoids of the non-degenerate quadric Q(4,q) of PG(4,q) have been studied since the end of the '80s. They are rare objects and, beside the classical example given by an elliptic quadric, only three classes are known for q odd, one class for $q$ even, and a sporadic example for $ìq=3^5. It is well known that to any ovoid of Q(4,q) a bivariate polynomial f(x,y) can be associated. In this paper we classify ovoids of Q(4,q) whose corresponding polynomial f(x,y) has 'low degree' compared with q, in particular deg(f)<(q/6.3)^(3/13)-1. Finally, as an application, {two classes} of permutation polynomials in characteristic 3 are obtained.

math.CO↗
arXiv2022

Non-minimum tensor rank Gabidulin codes

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}_{q}^{4\times4}$. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.

cs.IT↗
arXiv2021

Low c-differential uniformity for functions modified on subfields

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential uniformity and show several results. For example, we prove that given $β_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $δ_i$, and $L_i$ (specific linearized polynomials defined in terms of $β_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^nβ_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n δ_i$.

cs.IT↗
arXiv2021

Linear maximum rank distance codes of exceptional type

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 $\mathbb{F}_{q^n}$-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.

cs.IT↗
arXiv2021

Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Over the past few years, the codes $\mathcal{C}_{n-1}(n,q)$ arising from the incidence of points and hyperplanes in the projective space $\text{PG}(n,q)$ attracted a lot of attention. In particular, small weight codewords of $\mathcal{C}_{n-1}(n,q)$ are a topic of investigation. The main result of this work states that, if $q$ is large enough and not prime, a codeword having weight smaller than roughly $\frac{1}{2^{n-2}}q^{n-1}\sqrt{q}$ can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.

math.CO↗