Daniele Bartoli

Daniele Bartoli, Pantelimon Stanica

We study trivariate permutation polynomials over $\mathbb{F}_{2^{m}}$ extending two APN permutation families of Li--Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over $\mathbb{F}_{2^m}^*$. For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where $a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in $\mathbb{F}_{2^m}^*$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least \[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \] values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li--Kaleyski family. For the second family \[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \] we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\neq 6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\neq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on $\mathbb{F}_{2^{3m}}$.

Daniele Bartoli, Alessandro Giannoni, Giuseppe Marino et al.

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.

Daniele Bartoli, Giovanni Giuseppe Grimaldi, Pantelimon Stănică

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.