arXiv subjects

Alessandro Giannoni

Publications and source records attributed to Alessandro Giannoni.

11 recordsLinked to original sources

arXiv2026

F-sets of arbitrary finite width

Ferraguti and Micheli introduced the width of an $F$-set and conjectured that non-trivial $F$-sets of arbitrary width exist over every finite field. For $q\neq 2,3$, their constructions give examples of widths one and two. We prove that, for every $q\neq 2,3$ and every integer $r\geq 1$, there exists an infinite, non-trivial $F$-set in $\mathbb F_q[X]$ of width exactly $r$. Thus the finite-width part of their conjecture is settled over all such fields. The proof combines a bounded-core family of irreducible power substitutions with factorization results for $g(X^n)$, Dirichlet's theorem over $\mathbb F_q[X]$, and Kummer lifting. Core degree gives a uniform upper bound on the width, while parallel successor ladders give the required lower bound; a suitable tail of the nullity filtration then has the prescribed width.

math.NT↗

arXiv2026

A General Construction of Codes from Drinfeld Modules

We construct additive rank-metric and sum-rank-metric codes from Drinfeld modules by restricting bounded-degree morphisms to prime-to-characteristic torsion. For supersingular Drinfeld modules of rank $r$ in characteristic $\mathfrak{p}$ of degree $d$, the stabilization formula for morphism spaces yields rank-metric codes of $\mathbb{F}_q$-dimension $mrt-c$ and minimum distance $r-t+1$, where $c=r(r-1)(d-1)/2$. Simultaneous restriction to $\ell$ distinct degree-$m$ torsion modules gives additive sum-rank codes of the same dimension and minimum distance at least $\ell r-t+1$. Their normalized Singleton defects tend to zero, while in characteristic $(T)$ the module $ϕ_T=τ^r$ makes the defect vanish and produces an explicit MSRD family. We identify this family with a skew Chinese remainder theorem code supported on central skew polynomials and prove that its poly-skew weight is exactly $m$ times its sum-rank weight. This gives a specialized Singleton-type bound and a polynomial-time unique decoder up to the full sum-rank unique-decoding radius. We also derive a Welch-Berlekamp-type filter equation for the general supersingular sum-rank construction; it becomes an effective decoder whenever bases of the relevant morphism spaces and the restriction maps are computable.

math.NT↗

arXiv2026

The extremal gap for scattered $q$-systems

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 they coincide. Besides $m=h+1$ and $m=h+2$, equality occurs precisely when $h$ is even, $m=h+3$, and $k\equiv h/2\pmod{h+1}$. We also prove rigidity in the first two cases: every extremal system is equivalent to either $\mathbb{F}_q^k$ or a direct sum of elementary Gabidulin systems and $\mathbb{F}_q$-directions. For $h=m-3$ and $k=\ell(m-2)+s$, with $1\leq s<(m-2)/2$, the two bounds differ by one, apart from one boundary case. Starting from an arbitrary maximum scattered $q$-system, we construct extremal quasi-maximum $(m-3)$-scattered systems of rank $\ell m+s$. Every such extension preserves lower bounds on the generalized rank weights of the initial code. When an explicit independence condition holds, the construction also produces a nondegenerate dual pair of quasi-MRD codes. We parametrize these extensions by subspaces of a quotient space and derive a uniform lower bound on the number of inequivalent outputs. Known order-two families yield codes separated from every nontrivial direct sum under an explicit numerical condition, while higher-order families are separated from a specified order-one direct-sum class. We conclude with a fully explicit binary example.

math.CO↗

arXiv2026

Generalizing a family of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $ψ_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.

math.CO↗

arXiv2026

Thresholds for Tic-Tac-Toe on Finite Affine Spaces

We introduce an affine version of Tic-Tac-Toe played on the finite affine space $\mathbb{F}_q^m$. Two players alternately claim points, and the first player to occupy all points of an affine subspace of dimension $n$ wins. We call this the $(m,n)_q$-game. For fixed $n$ and $q$, we study how the outcome depends on the ambient dimension $m$. Using strategy stealing and a blocking-set interpretation, we show that every $(m,n)_q$-game is either a first-player win or a draw, and that the property of being a first-player win is monotone in $m$. This yields a threshold $T(n,q)$: the game is a draw for $m<T(n,q)$ and a first-player win for $m\ge T(n,q)$. We prove that this threshold is finite by applying the affine/vector-space Ramsey theorem of Graham, Leeb and Rothschild, and we obtain general lower bounds from the Erdős-Selfridge criterion for Maker-Breaker games. In the binary case, we give a direct Fourier-analytic argument, combined with an inductive lifting method, which shows that \[ T(n,2)\le 2^{n+1}. \] We also determine several small cases, including $T(1,q)=2$ for $q\in\{2,3,4\}$ and $T(2,2)=4$, and we prove geometric lower bounds from explicit pairing strategies, such as $T(n,q)\ge n+2$ for every $n\ge 2$. Our results place affine Tic-Tac-Toe at the interface of strong positional games, finite geometry and Ramsey theory for finite affine spaces.

math.CO↗

arXiv2026

An infinite family of non-extendable MRD codes

In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.

cs.IT↗

arXiv2025

Long QMDS additive code

We investigate additive codes, defined as $\mathbb{F}_q$-linear subspaces $C \subseteq \mathbb{F}_{q^h}^n$ of length $n$ and dimension $r$ over $\mathbb{F}_q$. An additive code is said to be of type $[n, r/h, d]_q^h$, where $d$ denotes the minimum Hamming distance and the normalized dimension $r/h$ may be fractional. A central object of interest is the class of quasi-MDS (QMDS) codes, those additive codes achieving the generalized Singleton bound: $$ d = n - \left\lceil \frac{r}{h} \right\rceil + 1. $$ In this work, we construct explicit families of additive QMDS codes whose lengths exceed those of the best-known $\mathbb{F}_{q^h}$-linear MDS codes which is $q^h+1$, and we will call these types of codes ``Long'' . By leveraging $\mathbb{F}_q$-linearity and geometric tools like partial spreads and dimensional dual arcs, we show that additive structures allow longer codes without sacrificing optimality in distance. We also examine dual codes and give conditions under which the QMDS property is preserved under duality.

math.CO↗

arXiv2025

Towards the classification of scattered binomials

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 \in \mathbb{F}_q^n \setminus \{ 0 \} $, whenever $ \frac{f(x)}{x} = \frac{f(y)}{y} $, it follows that $ \frac{x}{y} \in \mathbb{F}_q $. In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by $f(x) = δx^{q^s} + x^{q^{n-s}},$ and the one from Csajbók, Marino, Polverino, and Zanella (CMPZ), given by $f(x) = δx^{q^s} + x^{q^{s + n/2}},$ where $ n = 6 $ or $ n = 8 $. Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when $ q $ is sufficiently large and $ n $ is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in $\Fn$ when $n\leq8$ and $q$ is large enough.

math.CO↗

arXiv2024

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

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of $h$-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.

math.CO↗

arXiv2024

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

Scattered subspaces and $h$-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist, relatively few are known about $h$-scattered subspaces with $h\geq2$. In this paper, we establish the existence of maximum $2$-scattered $\F_q$-subspaces in $V(r,q^6)$ whenever $r\geq 3$, $r\ne 5$, and $q$ is an odd power of $2$. Additionally, we explore the corresponding MRD codes.

math.CO↗

arXiv2023

New scattered sequences of order 3

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 scattered sequences of order three is obtained. Equivalence issues are also considered.

math.CO↗