Ilaria Cardinali

Ilaria Cardinali, Luca Giuzzi

Let ${\mathcal C}(Ω)$ be the linear code arising from a projective system $Ω$ of $\mathrm{PG}(V).$ Consider the point-line geometry $Γ=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon Γ\rightarrow \mathrm{PG}(V)$ of $Γ.$ We show that the projective code obtained by taking as projective system $Ω:=\varepsilon(\mathcal{P})$ is minimal if the graph induced on the set $Γ\setminus\varepsilon^{-1}(H)$ by the collinearity graph of $Γ$ is connected for any hyperplane $H$ of $\mathrm{PG}(V)$. As an application, Grassmann codes, Segre codes, polar Grassmann codes of orthogonal, symplectic, hermitian type and codes arising from the point-hyperplane geometry of a projective space are minimal codes.

Ilaria Cardinali, Luca Giuzzi

Let $\barΓ$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $σ$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_σ$ of $\barΓ$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[ξ])\in \barΓ$ to the projective point represented by the pure tensor $x^σ\otimes ξ$, with $ξ(x)=0.$ In [I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry -- part I: the Segre embedding (Jun. 2025). arXiv:2506.21309, doi:10.48550/ARXIV.2506.21309] we focused on the case $σ=1$ and we studied the projective code arising from the projective system $Λ_1=\varepsilon_{1}(\barΓ).$ Here we focus on the case $σ\not=1$ and we investigate the linear code ${\mathcal C}(Λ_σ)$ arising from the projective system $Λ_σ=\varepsilon_σ(\barΓ).$ In particular, after having verified that $\mathcal{C}( Λ_σ)$ is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group. We also give a (geometrical) characterization of its minimum and second lowest weight codewords and determine its maximum weight when $q$ and $n$ are both odd.