Searcharxiv⌕ Search

arXiv subjects

Paolo Santonastaso

Publications and source records attributed to Paolo Santonastaso.

33 records · Page 2Linked to original sources
arXiv2023

On the minimum size of linear sets

Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $π$ in a canonical subgeometry. We obtain a tight lower bound on the size of any $\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $π$ is a hyperplane, and in the case that $π$ is a lower dimensional subspace.

math.CO↗
arXiv2023

On MSRD codes, h-designs and disjoint maximum scattered linear sets

In this paper we study geometric aspects of codes in the sum-rank metric. We establish the geometric description of generalised weights, and analyse the Delsarte and geometric dual operations. We establish a correspondence between maximum sum-rank distance codes and h-designs, extending the well-known correspondence between MDS codes and arcs in projective spaces and between MRD codes and h-scatttered subspaces. We use the geometric setting to construct new h-designs and new MSRD codes via new families of pairwise disjoint maximum scattered linear sets.

cs.IT↗
arXiv2023

Constructions and equivalence of Sidon spaces

Sidon spaces have been introduced by Bachoc, Serra and Zémor in 2017 as the $q$-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth, Raviv and Tamo in 2018, in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of $\mathbb{F}_{q^n}$, the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of $\mathbb{F}_{q^n}$ and finally the ones obtained as the kernel of subspace polynomials. In this paper we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples.

math.CO↗
arXiv2023

Maximum weight codewords of a linear rank metric code

Let $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear non-degenerate rank metric code with dimension $k$. In this paper we investigate the problem of determining the number $M(\mathcal{C})$ of codewords in $\mathcal{C}$ with maximum weight, that is $\min\{m,n\}$, and to characterize those with the maximum and the minimum values of $M(\mathcal{C})$.

cs.IT↗
arXiv2022

Divisible linear rank metric codes

A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ can be naturally embedded as a subspace of matrices in $\mathbb{F}_q^{em\times en}$ with the property that the rank of any of its matrix is a multiple of $e$. It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.

cs.IT↗
arXiv2022

Cones from maximum $h$-scattered linear sets and a stability result

This paper mainly focuses on cones whose basis is a maximum $h$-scattered linear set. We start by investigating the intersection sizes of such cones with the hyperplanes. Then we analyze two constructions of point sets with few intersection sizes with the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum $h$-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.

math.CO↗
arXiv2022

Clubs and their applications

Clubs of rank k are well-celebrated objects in finite geometries introduced by Fancsali and Sziklai in 2006. After the connection with a special type of arcs known as KM-arcs, they renewed their interest. This paper aims to study clubs of rank n in PG$(1,q^n)$. We provide a classification result for (n-2)-clubs of rank n, we analyze the $\mathrm{ΓL}(2,q^n)$-equivalence of the known subspaces defining clubs, for some of them the problem is then translated in determining whether or not certain scattered spaces are equivalent. Then we find a polynomial description of the known families of clubs via some linearized polynomials. Then we apply our results to the theory of blocking sets, KM-arcs, polynomials and rank metric codes, obtaining new constructions and classification results.

math.CO↗
arXiv2022

Classifications and constructions of minimum size linear sets

This paper aims to study linear sets of minimum size in the projective line, that is $\mathbb{F}_q$-linear sets of rank $k$ in $\mathrm{PG}(1,q^n)$ admitting one point of weight one and having size $q^{k-1}+1$. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for $k\leq 5$. In this paper we provide classification results for those $L_U$ admitting two points with complementary weights. We construct new examples and also study the related $\mathrm{ΓL}(2,q^n)$-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.

math.CO↗
arXiv2021

Two pointsets in $\mathrm{PG}(2,q^n)$ and the associated codes

In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set $L$. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing $L$ to be an $\mathbb{F}_q$-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the $Γ\mathrm{L}$-class of $L$ and the number of inequivalent codes we can construct starting from it.

math.CO↗
arXiv2021

The geometry of one-weight codes in the sum-rank metric

We provide a geometric characterization of $k$-dimensional $\mathbb{F}_{q^m}$-linear sum-rank metric codes as tuples of $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when $k=2$, they are one-weight, as in the Hamming-metric case. We then focus on constant rank-profile codes in the sum-rank metric, which are a special family of one weight-codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the $n$-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of $\mathrm{GL}(k,q^m)$. They turn out to be constant rank-profile - and hence one-weight - and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on $2$-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when $q=2$.

cs.IT↗
arXiv2021

Linear sets on the projective line with complementary weights

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021).

math.CO↗
arXiv2021

Linearized trinomials with maximum kernel

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $σ$ be a generator of $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. In this paper we provide closed formulas for the coefficients of a $σ$-trinomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the dimension of the kernel of $f$ equals its $σ$-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having $σ$-degree $3$ and $4$. Our techniques rely on the tools developed in [24]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8], to construct quasi-subfield polynomials and cyclic subspace codes, obtaining new explicit constructions to the conjecture posed in [37].

math.NT↗
arXiv2021

Extending two families of maximum rank distance codes

In this paper we provide a large family of rank-metric codes, which contains properly the codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are $\mathbb{F}_{q^{2t}}$-linear of dimension $2$ in the space of linearized polynomials over $\mathbb{F}_{q^{2t}}$, where $t$ is any integer greater than $2$, and we prove that they are maximum rank distance codes. For $t\ge 5$, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.

cs.IT↗
arXiv2021

On the list decodability of rank-metric codes containing Gabidulin codes

Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be efficiently list decoded for any radius $τ$, providing that $τ$ is large enough. Also, they proved that there are infinitely many choices of the parameters for which Gabidulin codes cannot be efficiently list decoded at all. Subsequently, in [41] these results have been extended to the family of generalized Gabidulin codes and to further family of MRD-codes. In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or not a polynomial-time list decoding algorithm exists. We detect several families of rank-metric codes containing a generalized Gabidulin code as subcode which cannot be efficiently list decoded for any radius large enough and families of rank-metric codes which cannot be efficiently list decoded. These results suggest that rank-metric codes which are $\mathbb{F}_{q^m}$-linear or that contains a (power of) generalized Gabidulin code cannot be efficiently list decoded for large values of the radius.

cs.IT↗