A. Giannoni

activity

20232026

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11 papers · 1 filter

math.CO2026

The weak cylinder conjecture for polynomial graphs of bounded degree

Daniele Bartoli, Alessandro Giannoni

Ball's weak cylinder conjecture asserts that a set of $p^{2}$ points in $AG (3, p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the…

math.CO2026

Maximum 2-scattered subspaces of $V (r, q^{6})$ for every $r \geq 3$

Daniele Bartoli, Francesco Ghiandoni, Alessandro Giannoni +2

We prove that for every prime power $q$ and every integer $r \geq 3$ there exists a maximum $2$ -scattered $F_{q}$ -subspace of $V (r, q^{6})$ , that is, a $2 r$ -dimensional $\mathbb…

math.CO2026

The extremal gap for scattered $q$ -systems

Alessandro Giannoni

Two general estimates govern the rank of an $h$ -scattered $q$ -system: the scattered upper bound and the lower bound for systems maximal under inclusion. We determine exactly when t…

math.CO2026

Thresholds for Tic-Tac-Toe on Finite Affine Spaces

Luca Bastioni, Alessandro Giannoni, Javier Lobillo-Olmedo

We introduce an affine version of Tic-Tac-Toe played on the finite affine space $F_{q}^{m}$ . Two players alternately claim points, and the first player to occupy all points of…

math.CO2026

Generalizing a family of scattered quadrinomials in $F_{q^{2 t}} [X]$

Alessandro Giannoni, Giovanni Giuseppe Grimaldi, Giovanni Longobardi +1

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $F_{q^{n}} [X]$ are known to exist for i…

math.CO2025

Long QMDS additive code

Daniele Bartoli, Alessandro Giannoni, Giuseppe Marino +1

We investigate additive codes, defined as $F_{q}$ -linear subspaces $C \subseteq F_{q^{h}}^{n}$ of length $n$ and dimension $r$ over $F_{q}$ . An additive code is…

math.CO2025

Towards the classification of scattered binomials

Daniele Bartoli, Francesco Ghiandoni, Alessandro Giannoni +1

Let $ q $ be a prime power and $ n $ an integer. An $ \mathbb{F}_q $-linearized polynomial $ f $ is said to be scattered if it satisfies the condition that for all \( x, y…

math.CO2024

A new infinite family of maximum $h$ -scattered $F_{q}$ -subspaces of $V (m (h + 1), q^{n})$ and associated MRD codes

Daniele Bartoli, Alessandro Giannoni, Giuseppe Marino

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within f…

math.CO2024

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

Daniele Bartoli, Francesco Ghiandoni, Alessandro Giannoni +1

Scattered subspaces and $h$ -scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While n…

math.CO2024

New scattered subspaces in higher dimensions

D. Bartoli, A. Giannoni, G. Marino

Over the past few decades, there has been extensive research on scattered subspaces, partly because of their link to MRD codes. These subspaces can be characterized using linearize…

math.CO2023

New scattered sequences of order 3

Daniele Bartoli, Alessandro Giannoni

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…

The weak cylinder conjecture for polynomial graphs of bounded degree

Maximum 2-scattered subspaces of V(r,q6) for every r≥3

The extremal gap for scattered q-systems

Thresholds for Tic-Tac-Toe on Finite Affine Spaces

Generalizing a family of scattered quadrinomials in Fq2t​[X]

Long QMDS additive code

Towards the classification of scattered binomials

A new infinite family of maximum h-scattered Fq​-subspaces of V(m(h+1),qn) and associated MRD codes

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

New scattered subspaces in higher dimensions

New scattered sequences of order 3

Maximum 2-scattered subspaces of $V (r, q^{6})$ for every $r \geq 3$

The extremal gap for scattered $q$ -systems

Generalizing a family of scattered quadrinomials in $F_{q^{2 t}} [X]$

A new infinite family of maximum $h$ -scattered $F_{q}$ -subspaces of $V (m (h + 1), q^{n})$ and associated MRD codes

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