Papers by Ferdinando Zullo
This paper presents encoding and decoding algorithms for several families of optimal rank metric ... more This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to $\lfloor(d-1)/2\rfloor$ in polynomial-time, where $d$ is the minimum distance of the code.
Lunardon and Polverino introduced in 2001 a new family of maximum scattered linear sets in $\math... more Lunardon and Polverino introduced in 2001 a new family of maximum scattered linear sets in $\mathrm{PG}(1,q^n)$ to construct linear minimal Rédei blocking sets. This family has been extended first by Lavrauw, Marino, Trombetti and Polverino in 2015 and then by Sheekey in 2016 in two different contexts (semifields and rank metric codes). These linear sets are called Lunardon-Polverino linear sets and this paper aims to determine its automorphism group, to solve the equivalence issue among Lunardon-Polverino linear sets and to establish the number of inequivalent linear sets of this family. We then elaborate on this number, providing explicit bounds and determining its asymptotics.
We investigate two fundamental questions intersecting coding theory and combinatorial geometry, w... more We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones...
Discrete Mathematics, 2022
Linear sets on the projective line have attracted a lot of attention because of their link with b... more 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).
Hermitian Rank-Metric Codes
Trends in Mathematics, 2021
Advances in Mathematics of Communications, 2022
In this paper we consider two pointsets in PG(2, q n) arising from a linear set L of rank n conta... more In this paper we consider two pointsets in PG(2, q n) arising from a linear set L of rank n contained in a line of 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 Fq-linear set with a short weight distribution, then the associated codes have few weights. We conclude the paper by providing a connection between the ΓL-class of L and the number of inequivalent codes we can construct starting from it.
ArXiv, 2021
Sidon spaces have been introduced by Bachoc, Serra and Zémor in 2017 in connection with the linea... more Sidon spaces have been introduced by Bachoc, Serra and Zémor in 2017 in connection with the linear analogue of Vosper’s Theorem. In this paper, we propose a generalization of this notion to sets of subspaces, which we call multi-Sidon space. We analyze their structures, provide examples and introduce a notion of equivalnce among them. Making use of these results, we study a class of linear sets in PG(r − 1, q) determined by r points and we investigate multi-orbit cyclic subspace codes.
Journal of Combinatorial Theory, Series A, 2022
In this paper we prove that the property of being scattered for a F q-linearized polynomial of sm... more In this paper we prove that the property of being scattered for a F q-linearized polynomial of small q-degree over a finite field F q n is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate r-fat polynomials, a natural generalization of scattered polynomials. An r-fat F q-linearized polynomial defines a linear set of rank n in the projective line of order q n with r points of weight larger than one. When r equals 1, the corresponding linear sets are called clubs, and they are related with a number of remarkable mathematical objects like KM-arcs, group divisible designs and rank metric codes. Using techniques on algebraic curves and global function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r > 0. In the case n ≤ 4, we completely determine the spectrum of values of r for which an r-fat polynomial exists. In the case n = 5, we provide a new family of 1-fat polynomials. Furthermore, we determine the values of r for which the so-called LP-polynomials are r-fat.
Designs, Codes and Cryptography, 2022
Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be ef... more 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 F q m-linear or that contains a (power of) generalized Gabidulin code cannot be efficiently list decoded for large values of the radius.
ArXiv, 2021
Maximum Hermitian rank metric codes were introduced by Schmidt in 2018 and in this paper we propo... more Maximum Hermitian rank metric codes were introduced by Schmidt in 2018 and in this paper we propose both interpolation-based encoding and decoding algorithms for this family of codes when the length and the minimum distance of the code are both odd.
The aim of this paper is to survey on the known results on maximum scattered linear sets and MRD-... more The aim of this paper is to survey on the known results on maximum scattered linear sets and MRD-codes. In particular, we investigate the link between these two areas. In "A new family of linear maximum rank distance codes" (2016) Sheekey showed how maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ define square MRD-codes. Later in "Maximum scattered linear sets and MRD-codes" (2017) maximum scattered linear sets in $\mathrm{PG}(r-1,q^n)$, $r>2$, were used to construct non square MRD-codes. Here, we point out a new relation regarding the other direction. We also provide an alternative proof of the well-known Blokhuis-Lavrauw's bound for the rank of maximum scattered linear sets shown in "Scattered spaces with respect to a spread in $\mathrm{PG}(n,q)$" (2000).
Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace... more Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $h$. For $h=1$ and $\dim_q U=rn/2$ these subspaces were first studied by Blokhuis and Lavrauw in 2000 and are called maximum scattered. They have been investigated intensively because of their relations with two-weight codes and strongly regular graphs. In the recent years maximum scattered and $n$-dimensional $(r-1)$-scattered $\mathbb{F}_q$-subspaces have been linked also to MRD-codes with a maximum idealiser. In this paper we prove the upper bound $rn/(h+1)$ for the dimension of $h$-scattered subspaces, $h>1$, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.
In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dim... more In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of $\mathbb{F}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1\pmod 5$, has been proved. In this paper we show that this family produces $\mathbb{F}_q$-linear MRD-codes of $\mathbb{F}_q^{6\times 6}$, with the same properties, also in the remaining $q$ odd cases, but not in the $q$ even case. These MRD-codes are not equivalent to the previously known MRD-codes. We also prove that the corresponding maximum scattered $\mathbb{F}_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not $\mathrm{P}\Gamma\mathrm{L}(2,q^6)$-equivalent to any previously known scattered linear set.
Finite Fields and Their Applications, 2022
Scattered polynomials over a finite field F q n have been introduced by Sheekey in 2016, and a ce... more Scattered polynomials over a finite field F q n have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-q t-partially scattered and R-q t-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-q tpartially scattered and R-q t-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is is the hardest to be preserved is the L-q t-partially scattered one. On the one hand, we are able to extend the classification results of exceptional scattered polynomials to exceptional L-q t-partially scattered polynomials. On the other hand, the R-q t-partially scattered property seems more stable. We present a large family of R-q t-partially scattered polynomials, containing examples of exceptional R-q t-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. In order to detect new examples of polynomials which are R-q t-partially scattered, we introduce two different notions of equivalence preserving this property and concerning natural actions of the groups ΓL(2, q n) and ΓL(2n/t, q t). In particular, our family contains many examples of inequivalent polynomials, and geometric arguments are used to determine the equivalence classes under the action of ΓL(2n/t, q t).
Designs, Codes and Cryptography, 2021
After a seminal paper by Shekeey (2016), a connection between maximum h-scattered F q-subspaces o... more After a seminal paper by Shekeey (2016), a connection between maximum h-scattered F q-subspaces of V (r, q n) and maximum rank distance (MRD) codes has been established in the extremal cases h = 1 and h = r − 1. In this paper, we propose a connection for any h ∈ {1,. .. , r−1}, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.
Combinatorica, 2021
Let V be an r-dimensional F q n-vector space. We call an F q-subspace U of V h-scattered if U mee... more Let V be an r-dimensional F q n-vector space. We call an F q-subspace U of V h-scattered if U meets the h-dimensional F q n-subspaces of V in F q-subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that dim Fq U ≤ rn/2 when U is 1-scattered. Subspaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to rn/2-dimensional 1-scattered subspaces and to n-dimensional (r − 1)scattered subspaces. In this paper we prove the upper bound rn/(h+1) for the dimension of h-scattered subspaces, h > 1, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.
Finite Fields and Their Applications, 2021
Every linear set in a Galois space is the projection of a subgeometry, and most known characteriz... more Every linear set in a Galois space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept of linear sets of h-pseudoregulus type, which turns out to be projected from the span of an arbitrary number of director spaces of a Desarguesian spread of a subgeometry. Among these linear sets, we characterize those which are h-scattered and solve the equivalence problem between them; a key role is played by an algebraic tool recently introduced in the literature and known as Moore exponent set. As a byproduct, we classify asymptotically h-scattered linear sets of h-pseudoregulus type.
Journal of Pure and Applied Algebra, 2022
Linearized polynomials have attracted a lot of attention because of their applications in both ge... more 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 Gal(F q n : F q). In this paper we provide closed formulas for the coefficients of a σ-trinomial f over 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].
Ars Mathematica Contemporanea, 2020
We generalize the example of linear set presented by the last two authors in "Vertex properties o... more We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of PG(1, q n)" (2019) to a more general family, proving that such linear sets are maximum scattered when q is odd and, apart from a special case, they are are new. This solves an open problem posed in "Vertex properties of maximum scattered linear sets of PG(1, q n)" (2019). As a consequence of Sheekey's results in "A new family of linear maximum rank distance codes" (2016), this family yields to new MRD-codes with parameters (6, 6, q; 5).
Advances in Mathematics of Communications, 2019
In this article we present a class of codes with few weights arising from special type of linear ... more In this article we present a class of codes with few weights arising from special type of linear sets. We explicitly show the weights of such codes, their weight enumerator and possible choices for their generator matrices. In particular, our construction yields also to linear codes with three weights and, in some cases, to almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.
Uploads
Papers by Ferdinando Zullo