Quasi-optimal cyclic orbit codes Chiara Castello, Heide Gluesing-Luerssen, Olga Polverino, Ferdinando Zullo Designs Codes and Cryptography, 2026 We focus on two aspects of cyclic orbit codes: invariants under equivalence and quasi-optimality. Regarding the first aspect, we establish a connection between the cyclic orbit code generated by a subspace U of $${{\\mathbb {F}}}_{q^n}$$ F q n and the associated linear set $$L_{U\\times U}$$ L U × U . Relating the size of the linear set to the number of fractions formed by the elements of U allows us to derive new bounds on the parameters of the cyclic orbit code. In the second part, we study a particular family of (quasi-)optimal cyclic orbit codes. With the aid of these codes we establish the existence of quasi-optimal codes in even-dimensional vector spaces over finite fields of any characteristic. Finally, for the particular code family we determine the automorphism groups in various general linear group, depending on the assumed ground field, and their orbits under the Galois group over the prime field.
Rank-metric codes over arbitrary fields: Bounds and constructions Alessandro Neri, Ferdinando Zullo Art of Discrete and Applied Mathematics, 2026 Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appeared in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions.
REPRESENTABILITY OF THE DIRECT SUM OF UNIFORM q-MATROIDS Gianira N. Alfarano, Relinde Jurrius, Alessandro Neri, Ferdinando Zullo Combinatorial Theory, 2026 There are many similarities between the theories of matroids and \(q\)-matroids. However, when dealing with the direct sum of \(q\)-matroids many differences arise. Most notably, it has recently been shown that the direct sum of representable \(q\)-matroids is not necessarily representable. In this work, we focus on the direct sum of uniform \(q\)-matroids. Using algebraic and geometric tools, together with the notion of cyclic flats of \(q\)-matroids, we show that this is always representable, by providing a representation over a sufficiently large field.Mathematics Subject Classifications: 05B35, 94B05, 51E20Keywords: \(q\)-matroids, representability, evasive subspaces, rank-metric codes, linear sets
On the stabilizer of the graph of linear functions over finite fields Valentino Smaldore, Corrado Zanella, Ferdinando Zullo Forum Mathematicum, 2025 In this paper we will study the action of 𝔽 q n 2 × 2 {\\mathbb{F}_{q^{n}}^{2\\times 2}} on the graph of an 𝔽 q {{{\\mathbb{F}}_{q}}} -linear function of 𝔽 q n {{{\\mathbb{F}}_{q^{n}}}} to itself. In particular, we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered.
Neural network for archaeological glyph detection Serena Crisci, Valentina De Simone, Andrea Diana, Ferdinando Zullo Intelligent Systems with Applications, 2025 The increasing availability of visual data in fields such as archaeology has highlighted the need for automated image analysis tools. Ancient rock engravings, such as those in the Neolithic Domus de Janas tombs of Sardinia, are crucial cultural artifacts. However, their study is hindered by environmental degradation and the limitations of traditional analysis methods. This paper introduces a novel approach that employs a preprocessing method to isolate glyphs from their backgrounds, reducing the impact of wear and distortions caused by environmental factors such as lighting. Convolutional neural networks are then used to enhance the classification of glyphs in the preprocessed archaeological images. The refined data are processed using AlexNet, GoogLeNet, and EfficientNet neural networks, each trained to classify glyphs into distinct categories and to detect their geometric features. This method offers a more efficient and accurate way to analyze and preserve these cultural artifacts.
Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes Ferruh Özbudak, Paolo Santonastaso, Ferdinando Zullo Cryptography and Communications, 2025 We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain new classes of optical orthogonal codes with different parameters.
Clubs and Their Applications Vito Napolitano, Olga Polverino, Paolo Santonastaso, Ferdinando Zullo SIAM Journal on Applied Algebra and Geometry, 2024
Two-Weight Rank-Metric Codes Ferdinando Zullo, Olga Polverino, Paolo Santonastaso, John Sheekey IEEE International Symposium on Information Theory Proceedings, 2024
