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Stefano Lia

Publications and source records attributed to Stefano Lia.

13 recordsLinked to original sources
arXiv2026

Generalising Latin square orthogonality and Frobenius-K\"onig with alternating sign matrices

The theory of Latin squares has a long history. While the objects themselves appeared earlier, the study of their general mathematical theory dates back to Euler in the 18th century. Latin squares can be interpreted as 3-dimensional permutation hypermatrices, and alternating sign matrices often arise as a natural generalisation of permutation matrices. In 2018, Brualdi and Dahl introduced a generalisation of classical Latin squares using alternating sign hypermatrices. Inspired by their definition, we develop the theory of Italian squares, a related generalisation of Latin squares, together with a notion of orthogonality that resolves an inconsistency in the definition of Brualdi and Dahl. Building on classical questions from Latin square theory, we obtain results including upper bounds on the maximal size of a pairwise orthogonal set, conditions for the existence of an orthogonal mate, infinite families of orthogonal pairs, and transversals. As part of our exploration of alternating sign matrices, we also prove a Frobenius-K\"onig type result for a class of $(0,\pm1)$-matrices.

math.CO↗
arXiv2026

On the existence of a morphism between certain Artin-Schreier curves

It is well known that, given two curves $\mathcal{X}: y^p+cy=x^m$ and $\mathcal{Y}:y^p+cy=x^n$, defined over $\F_p$, if $n$ divides $m$ then there exists a nonconstant morphism $\mathcal{X} \longrightarrow \mathcal{Y}$. In this paper we are interested in studying whether the converse of this statement is true, i.e., if there exists a morphism $\mathcal{X} \longrightarrow\mathcal{Y}$ then must it be true that $n$ divides $m$? In particular, we consider the case when $m=p^{k}+1$ and $n=p^\ell+1$. We prove that the converse is true under certain hypotheses. We deal with both the cases of Galois morphisms and non-Galois morphisms.

math.AG↗
arXiv2026

Computational Explorations on Semifields

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.

cs.SC↗
arXiv2025

Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $\Sigma$ of $\mathrm{PG}(4,q^5)$ from a plane $\Gamma$ external to the secant variety to $\Sigma$. The pair $(\Gamma,\Sigma)$ will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in $\mathrm{PG}(1,q^5)$, namely those of pseudoregulus and LP type, have been characterized in the literature by B. Csajb\'{o}k, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. Let $(\Gamma,\Sigma)$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$, let $\sigma$ be a generator of $\mathbb{G}=\mathrm{P}\Gamma \mathrm{L}(5,q^5)_\Sigma$, and $A=\Gamma\cap\Gamma^{\sigma^4}$, $B=\Gamma\cap\Gamma^{\sigma^3}$. If $A$ and $B$ are not both points, then the projected linear set is of pseudoregulus type. Then, suppose that they are points. The rank of a point $X$ is the vectorial dimension of the span of the orbit of $X$ under the action of $\mathbb{G}$. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points $A$ and $B$ has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that $\mathrm{rk} A=\mathrm{rk} B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\leq 25$, no new maximum scattered linear set exists.

math.CO↗
arXiv2025

The Random Variables of the DNA Coverage Depth Problem

DNA data storage systems encode digital data into DNA strands, enabling dense and durable storage. Efficient data retrieval depends on coverage depth, a key performance metric. We study the random access coverage depth problem and focus on minimizing the expected number of reads needed to recover information strands encoded via a linear code. We compute the asymptotic performance of a recently proposed code construction, establishing and refining a conjecture in the field by giving two independent proofs. We also analyze a geometric code construction based on balanced quasi-arcs and optimize its parameters. Finally, we investigate the full distribution of the random variables that arise in the coverage depth problem, of which the traditionally studied expectation is just the first moment. This allows us to distinguish between code constructions that, at first glance, may appear to behave identically.

cs.IT↗
arXiv2024

On the geometry of tensor products over finite fields

In this paper we study finite dimensional algebras, in particular finite semifields, through their correspondence with nonsingular threefold tensors. We introduce a alternative embedding of the tensor product space into a projective space. This model allows us to understand tensors and their contractions in a new geometric way, relating the contraction of a tensor with a natural subspace of a subgeometry. This leads us to new results on invariants and classifications of tensors and algebras and on nonsingular fourfold tensors. A detailed study of the geometry of this setup for the case of the threefold tensor power of a vector space of dimension two over a finite field surprisingly leads to a new construction of quasi-hermitian varieties in $\mathrm{PG}(3,q^2)$.

math.CO↗
arXiv2024

Short rank-metric codes and scattered subspaces

By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new class of codes with parameters $[m+2,3,m-2]_{q^m/q}$ for infinite values of $q$ and $m \geq 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

cs.IT↗
arXiv2024

A note on strong blocking sets and higgledy-piggledy sets of lines

This paper studies {\em strong blocking sets} in the $N$-dimensional finite projective space $\mathrm{PG}(N,q)$. We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the size of a strong blocking set in $\mathrm{PG}(N,q)$. Our second main result shows that, for $q>\frac{2}{ln(2)}(N+1)$, there exists a subset of $2N-2$ lines of a Desarguesian line spread in $\mathrm{PG}(N,q)$, $N$ odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong blocking set of size $(2N-2)(q+1)$.

math.CO↗
arXiv2023

On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

The complete classification of the orbits on subspaces under the action of the projective stabiliser of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of $\PGL(2,q^2)$ (and $\PSL(2,q^2)$) arising from the Hermitian Veronese curve in $\PG(3, q^2)$, a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermitian surface.

math.CO↗
arXiv2021

A generalization of Bring's curve in any characteristic

Let $p\ge 7$ be a prime, and $m\ge 5$ an integer. A natural generalization of Bring's curve valid over any field $\mathbb{K}$ of zero characteristic or positive characteristic $p$, is the algebraic variety $V$ of $\textrm{PG}(m-1,\mathbb{K})$ which is the complete intersection of the projective algebraic hypersurfaces of homogeneous equations $x_1^k+\cdots +x_m^{k}=0$ with $1\leq k\leq m-2$. In positive characteristic, we also assume $m\le p-1$. Up to a change of coordinates in $\textrm{PG}(m-1,\mathbb{K})$, we show that $V$ is a projective, absolutely irreducible, non-singular curve of $\textrm{PG}(m-2,\mathbb{K})$ with degree $(m-2)!$, genus $\mathfrak{g}= \frac{1}{4} ((m-2)(m-3)-4)(m-2)!+1$, and tame automorphism group $G$ isomorphic to $\textrm{Sym}_m$. We compute the genera of the quotient curves of $V$ with respect to the stabilizers of one or more coordinates under the action of $G$. In positive characteristic, the two extremal cases, $m=5$ and $m=p-1$ are investigated further. For $m=5$, we show that there exist infinitely many primes $p$ such that $V$ is $\mathbb{F}_{p^2}$-maximal curve of genus $4$. The smallest such primes are $29,59,149,239,839$. For $m=p-1$ we prove that $V$ has as many as $(p-2)!$ points over $\mathbb{F}_p$ and has no further points over $\mathbb{F}_{p^2}$. We also point out a connection with previous work of R\'edei about the famous Minkowski conjecture proven by Haj\'os (1941), as well as with a more recent result of Rodr\'iguez Villegas, Voloch and Zagier (2001) on plane curves attaining the St\"ohr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced by Conca, Krattenthaler and Watanabe (2009).

math.AG↗
arXiv2020

Curves with more than one inner Galois point

Let $\mathcal{C}$ be an irreducible plane curve of $\text{PG}(2,\mathbb{K})$ where $\mathbb{K}$ is an algebraically closed field of characteristic $p\geq 0$. A point $Q\in \mathcal{C}$ is an inner Galois point for $\mathcal{C}$ if the projection $π_Q$ from $Q$ is Galois. Assume that $\mathcal{C}$ has two different inner Galois points $Q_1$ and $Q_2$, both simple. Let $G_1$ and $G_2$ be the respective Galois groups. Under the assumption that $G_i$ fixes $Q_i$, for $i=1,2$, we provide a complete classification of $G=\langle G_1,G_2 \rangle$ and we exhibit a curve for each such $G$. Our proof relies on deeper results from group theory.

math.AG↗
arXiv2018

An $\mathbb{F}_{p^2}$-maximal Wiman's sextic and its automorphisms

In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied as a curve defined over a finite field $\mathbb{F}_{p^2}$ where $p$ is a prime, and necessary and sufficient conditions for its maximality over $\mathbb{F}_{p^2}$ were obtained. In this paper we first show that the result of Wiman concerning the automorphism group of $\mathcal{W}$ holds also over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$, provided that $p \geq 7$. For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2$ is not irreducible over $\mathbb{K}$, while for $p=5$ the curve $\mathcal{W}$ is rational and $Aut(\mathcal{W}) \cong PGL(2,\mathbb{K})$. We also show that the $\mathbb{F}_{19^2}$-maximal Wiman's sextic $\mathcal{W}$ is not Galois covered by the Hermitian curve $\mathcal{H}_{19}$ over $\mathbb{F}_{19^2}$.

math.AG↗