arXiv subjects

Giovanni Giuseppe Grimaldi

Publications and source records attributed to Giovanni Giuseppe Grimaldi.

10 recordsLinked to original sources

arXiv2026

Breaking linearity in the sum-rank metric: MSRD codes from switched $σ$-rational normal curves

We construct a family of scalar-closed, non-additive maximum sum-rank distance codes (MSRD) by switching norm components of a $σ$-rational normal curve and lifting the resulting point set through a linearized Reed--Solomon syndrome map. We characterize the admissible switches by a multiplicative stability condition on the norm classes. In particular, for $1 \leq n_i \leq m$, $N=n_1+\ldots+n_{\ell}$ and $2\leqδ\leq N-1$, every partition of $\mathbb{F}_q^*$ satisfying a specific condition, referred to as Condition $(\diamond)$, yields a code in $\mathbb{F}_{q^m}^{N}$ of size $q^{m(N-δ+1)}$ and minimum sum-rank distance $δ$; the code is non-additive whenever a curve component is retained. For $\ell\geq2$ these appear to be the first non-additive MSRD codes in the literature, all previously known families with more than one block being $\mathbb{F}_q$-linear. Moreover, every code of the family has the same sum-rank weight distribution as an $\mathbb{F}_{q^m}$-linear MSRD code with the same parameters, although the geometry of the switched set distinguishes it from linearized Reed--Solomon codes. In the single-block case, an explicit rank isometry identifies the construction with the cone codes of Durante, Grimaldi and Longobardi.

cs.IT↗

arXiv2026

On the hull of linearized polynomial codes

Motivated by entanglement-assisted quantum error-correcting codes, where the hull dimension determines the number of required pre-shared entangled pairs, we study hulls of two families of $\mathbb{F}_q$-linear codes defined by $q$-polynomial operators over $\mathbb{F}_{q^m}$. Our main tool is a unified Gram-matrix method. For image codes $\mathcal{C}(\boldsymbolα)=\operatorname{im}Φ_{\boldsymbolα}$, with $Φ_{\boldsymbolα}=\sum_iα_iF_i$, we prove the master hull--rank formula $\dim\operatorname{Hull}(\mathcal{C}(\boldsymbolα))=\operatorname{rank}(Φ_{\boldsymbolα})-\operatorname{rank}(G(\boldsymbolα))$, where $G(\boldsymbolα)$ is the associated Gram matrix over $\mathbb{F}_q$. Specializing to $C_{λ,μ}=\operatorname{im}(λx+μL(x))$, we obtain a quadratic Gram pencil $λ^2G_0+λμG_1+μ^2G_2$ whose determinant describes the LCD locus in $\mathbb{P}^1(\mathbb{F}_q)$. We also treat $\mathbb{F}_{q^m}$-linear rank-distance codes $\mathcal{C}=\langle X,F_1,\ldots,F_k\rangle_{\mathbb{F}_{q^m}}$ with the Delsarte inner product, where a $k\times k$ Gram matrix over $\mathbb{F}_{q^m}$ determines the hull dimension. For $L(X)=X^{q^k}$, with $d=\gcd(k,m)$, the resulting circulant Gram matrices yield a closed-form discriminant and a complete classification in three of the four bijectivity configurations over $\mathbb{P}^1(\mathbb{F}_{q^m})$. In the remaining case, the hull dimension equals $δ=\dim_{\mathbb{F}_q}(\operatorname{im}ϕ_{λ,μ}\cap\kerϕ_{λ,μ}^{\dagger})$, and the extremal condition $δ=d$ is characterized by an explicit trace-isotropy criterion. We conclude with an exact count of LCD and non-LCD points, showing that the LCD density tends to $1$ as $q\to\infty$, together with a worked example over $\mathbb{F}_{64}$ and a SageMath verification.

cs.IT↗

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

Homomorphic encryption schemes based on coding theory and polynomials

Homomorphic encryption is a powerful cryptographic tool that enables secure computations on the private data. It evaluates any function for any operation securely on the encrypted data without knowing its corresponding plaintext. For original data $p$, $c$ denotes the ciphertext of the original plaintext $p$, i.e. $c = Encrypt_k(p)$. This is crucial for any sensitive application running in the Cloud, because we must protect data privacy even in the case when the server has falled victim to a cyber attack. The encryption scheme $Encrypt_k$ is said to be homomorphic with respect to some set of operations $\mathcal{O}$, if for any operation $\circ \in \mathcal{O}$ one can compute $Encrypt_k(p_1 \circ p_2)$ from $Encrypt_k(p_1) \circ Encrypt_k(p_2)$. Those schemes come in three forms: somewhat, partially and fully homomorphic. In this survey, we present the state of art of the known homomorphic encryption schemes based on coding theory and polynomials.

cs.CR↗

arXiv2026

On the Classification of Dillon's APN Hexanomials

We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family.

math.NT↗

arXiv2025

Ovoids of $Q^+(7,q)$ of low-degree

Ovoids of the hyperbolic quadric $Q^+(7,q)$ of $\mathrm{PG}(7,q)$ have been extensively studied over the past 40 years, partly due to their connections with other combinatorial objects. It is well known that the points of an ovoid of $Q^+(7,q)$ can be parametrized by three polynomials $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$. In this paper, we classify ovoids of $Q^+(7,q)$ of low degree, specifically under the assumption that $f_1(X,Y,Z)$, $f_2(X,Y,Z)$, $f_3(X,Y,Z)$ have degree at most 3. Our approach relies on the analysis of an algebraic hypersurface associated with the ovoid.

math.CO↗

arXiv2024

Non-linear MRD codes from cones over exterior sets

By using the notion of $d$-embedding $Γ$ of a (canonical) subgeometry $Σ$ and of exterior set with respect to the $h$-secant variety $Ω_{h}(\mathcal{A})$ of a subset $\mathcal{A}$, $ 0 \leq h \leq n-1$, in the finite projective space $\mathrm{PG}(n-1,q^n)$, $n \geq 3$, in this article we construct a class of non-linear $(n,n,q;d)$-MRD codes for any $ 2 \leq d \leq n-1$. A code $\mathcal{C}_{σ,T}$ of this class, where $1\in T \subset \mathbb{F}_q^*$ and $σ$ is a generator of $\mathrm{Gal}(\mathbb{F}_{q^n}|\mathbb{F}_q)$, arises from a cone of $\mathrm{PG}(n-1,q^n)$ with vertex an $(n-d-2)$-dimensional subspace over a maximum exterior set $\mathcal{E}$ with respect to $Ω_{d-2}(Γ)$. We prove that the codes introduced in [Cossidente, A., Marino, G., Pavese, F.: Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597--609 (2016); Durante, N., Siciliano, A.: Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries. Electron. J. Comb. (2017); Donati, G., Durante, N.: A generalization of the normal rational curve in $\mathrm{PG}(d,q^n)$ and its associated non-linear MRD codes. Des. Codes Cryptogr. 86, 1175--1184 (2018)] are appropriate punctured ones of $\mathcal{C}_{σ,T}$ and solve completely the inequivalence issue for this class showing that $\mathcal{C}_{σ,T}$ is neither equivalent nor adjointly equivalent to the non-linear MRD code $\mathcal{C}_{n,k,σ,I}$, $I \subseteq \mathbb{F}_q$, obtained in [Otal, K., Özbudak, F.: Some new non-additive maximum rank distance codes. Finite Fields and Their Applications 50, 293--303 (2018).].

cs.IT↗

arXiv2024

A geometric characterization of known maximum scattered linear sets of $\mathrm{PG}(1,q^n)$

An $\mathbb{F}_q$- linear set $L=L_U$ of $Λ=\mathrm{PG}(V, \mathbb{F}_{q^n}) \cong \mathrm{PG}(r-1,q^n)$ is a set of points defined by non-zero vectors of an $\mathbb{F}_q$-subspace $U$ of $V$. The integer $\dim_{\mathbb{F}_q} U$ is called the rank of $L$. In [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004)], it was proven that any $\mathbb{F}_q$-linear set $L$ of $Λ$ of rank $u$ such that $\langle L \rangle=Λ$ is either a canonical subgeometry of $Λ$ or there are a $(u-r-1)$-dimensional subspace $Γ$ of $\mathrm{PG}(u-1,q^n) \supset Λ$ disjoint from $Λ$ and a canonical subgeometry $Σ\cong \mathrm{PG}(u-1,q)$ disjoint from $Γ$ such that $L$ is the projection of $Σ$ from $Γ$ onto $Λ$. The subspace $Γ$ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex $Γ$ for a peculiar class of linear sets of rank $u = n(r - 1)$ in $\mathrm{PG}(r - 1, q^n)$ called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line $\mathrm{PG}(1,q^n)$ introduced from 2018 onward, by means of certain properties of their projection vertices, as done in [B. Csajbók, C. Zanella: On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1, q^t)$, Finite Fields Appl. 41 (2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1, q^n)$. Discrete Math. 343(5) (2020)].

math.CO↗

arXiv2024

On $m$-ovoids of $Q^+(7,q)$ with $q$ odd

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of $\pg(3,q)$ that have as many secants to an elliptic quadric as possible.

math.CO↗

arXiv2023

On the classification of low degree ovoids of $Q^+(5,q)$

Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a three dimensional elliptic quadric (corresponding to the regular spread of $\mathrm{PG}(3,q)$) many other classes of examples are known. First of all the other examples (beside the elliptic quadric) of ovoids of $Q(4,q)$ give also examples of ovoids of $Q^+(5,q)$. Another important class of ovoids of $Q^+(5,q)$ is given by the ones associated to a flock of a three dimensional quadratic cone. To every ovoid of $Q^+(5,q)$ two bivariate polynomials $f_1(x,y)$ and $f_2(x,y)$ can be associated. In this paper, we classify ovoids of $Q^+(5,q)$ such that $f_1(x,y)=y+g(x)$ and $\max\{deg(f_1),deg(f_2)\}<(\frac{1}{6.3}q)^{\frac{3}{13}}-1$, that is $f_1(x,y)$ and $f_2(x,y)$ have "low degree" compared with $q$.

math.CO↗