Ferdinando Zullo
@unicampania.it
Department of Mathematics and Physics
University of Campania
EDUCATION
Bachelor, Master and PhD at University of Campania "Luigi Vanvitelli"
RESEARCH, TEACHING, or OTHER INTERESTS
Discrete Mathematics and Combinatorics, Theoretical Computer Science
Scopus Publications
Scopus Publications
Martino Borello and Ferdinando Zullo
Wiley
AbstractThe main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum‐rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank‐metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated Ashikhmin–Barg condition is proved and used to ensure the minimality of certain constructions.
Chiara Castello, Olga Polverino, Paolo Santonastaso, and Ferdinando Zullo
Springer Science and Business Media LLC
AbstractSidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017), 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 $$\\mathbb {F}_{q^n}$$ F q n , the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of $$\\mathbb {F}_{q^n}$$ 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.
Vito Napolitano, Olga Polverino, Paolo Santonastaso, and Ferdinando Zullo
Elsevier BV
Sam Adriaensen, Jonathan Mannaert, Paolo Santonastaso, and Ferdinando Zullo
Elsevier BV
Olga Polverino, Paolo Santonastaso, John Sheekey, and Ferdinando Zullo
Institute of Electrical and Electronics Engineers (IEEE)
A subspace of matrices in <inline-formula> <tex-math notation="LaTeX">${\\mathbb F}_{q^{e}}^{m\\times n}$ </tex-math></inline-formula> can be naturally embedded as a subspace of matrices in <inline-formula> <tex-math notation="LaTeX">${\\mathbb F}_{q}^{em\\times en}$ </tex-math></inline-formula> with the property that the rank of any of its matrix is a multiple of <inline-formula> <tex-math notation="LaTeX">$e$ </tex-math></inline-formula>. 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.
Daniele Bartoli, Giovanni Zini, and Ferdinando Zullo
Institute of Electrical and Electronics Engineers (IEEE)
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.
Wei Tang, Yue Zhou, and Ferdinando Zullo
Elsevier BV
Vito Napolitano, , and Ferdinando Zullo
American Institute of Mathematical Sciences (AIMS)
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.
Ferdinando Zullo
Elsevier BV
Vito Napolitano, , Olga Polverino, Paolo Santonastaso, and Ferdinando Zullo
American Institute of Mathematical Sciences (AIMS)
<p style='text-indent:20px;'>In this paper we consider two pointsets in <inline-formula><tex-math id="M2">\\begin{document}$ \\mathrm{PG}(2,q^n) $\\end{document}</tex-math></inline-formula> arising from a linear set <inline-formula><tex-math id="M3">\\begin{document}$ L $\\end{document}</tex-math></inline-formula> of rank <inline-formula><tex-math id="M4">\\begin{document}$ n $\\end{document}</tex-math></inline-formula> contained in a line of <inline-formula><tex-math id="M5">\\begin{document}$ \\mathrm{PG}(2,q^n) $\\end{document}</tex-math></inline-formula>: the first one is a linear blocking set of Rédei type, the second one extends the construction of translation KM-arcs. We point out that their intersections pattern with lines is related to the weight distribution of the considered linear set <inline-formula><tex-math id="M6">\\begin{document}$ L $\\end{document}</tex-math></inline-formula>. We then consider the Hamming metric codes associated with both these constructions, for which we can completely describe their weight distributions. By choosing <inline-formula><tex-math id="M7">\\begin{document}$ L $\\end{document}</tex-math></inline-formula> to be an <inline-formula><tex-math id="M8">\\begin{document}$ {\\mathbb F}_{q} $\\end{document}</tex-math></inline-formula>-linear set with a <i>short</i> weight distribution, then the associated codes have <i>few weights</i>. We conclude the paper by providing a connection between the <inline-formula><tex-math id="M9">\\begin{document}$ \\Gamma\\mathrm{L} $\\end{document}</tex-math></inline-formula>-class of <inline-formula><tex-math id="M10">\\begin{document}$ L $\\end{document}</tex-math></inline-formula> and the number of inequivalent codes we can construct starting from it.</p>
Alessandro Neri, Paolo Santonastaso, and Ferdinando Zullo
Elsevier BV
Anina Gruica, Alberto Ravagnani, John Sheekey, and Ferdinando Zullo
Society for Industrial & Applied Mathematics (SIAM)
Wrya K. Kadir, Chunlei Li, and Ferdinando Zullo
Springer Science and Business Media LLC
AbstractThis 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 ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code.
Daniele Bartoli, Giovanni Zini, and Ferdinando Zullo
Elsevier BV
Alessandro Neri, Paolo Santonastaso, and Ferdinando Zullo
Elsevier BV
Vito Napolitano, Olga Polverino, Paolo Santonastaso, and Ferdinando Zullo
Elsevier BV
Daniele Bartoli, Giacomo Micheli, Giovanni Zini, and Ferdinando Zullo
Elsevier BV
Daniele Bartoli, Marco Calderini, Olga Polverino, and Ferdinando Zullo
Elsevier BV
Paolo Santonastaso and Ferdinando Zullo
Elsevier BV
Paolo Santonastaso and Ferdinando Zullo
Springer Science and Business Media LLC
Daniele Bartoli, Giovanni Zini, and Ferdinando Zullo
Elsevier BV
Giovanni Zini and Ferdinando Zullo
Springer Science and Business Media LLC
Rocco Trombetti and Ferdinando Zullo
Springer Science and Business Media LLC
Wrya K. Kadir, Chunlei Li, and Ferdinando Zullo
IEEE
In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.
Giovanni Zini and Ferdinando Zullo
Elsevier BV