Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding
Ilaria Cardinali, Luca Giuzzi
TL;DR
The paper constructs a linear code from the Segre-embedded point-hyperplane geometry by restricting to the rank-1, zero-trace subvariety $\\Lambda_1$ and studies the resulting code $\\mathcal{C}(\\Lambda_1)$. It determines the code’s length $N_1$, dimension $k_1$, and minimum distance $d_1$, and provides the full weight spectrum and automorphism group $\\mathrm{PGL}(n+1,q)\cdot\mathbb{F}_q^{\star}$. A tight geometric interpretation is given for the minimum, second-minimum, and various maximum-weight codewords in terms of hyperplanes of the point-hyperplane geometry $\\bar{\\Gamma}$, including quasi-singular, singular, and spread-type hyperplanes. The work also establishes an encoding approach without an explicit generator matrix and proves that $d_1/N_1$ remains asymptotically near 1, even though the rate vanishes, highlighting the code’s strong asymptotic distance properties. A follow-up study extends these results to twisted variants of the Segre embedding and further exploits hyperplane geometry to refine weight classifications.</nobr>
Abstract
Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $Λ$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $Λ_{1}$ of $Λ$ represented by the pure tensors $x\otimes ξ$ with $x\in V$ and $ξ\in V^*$ such that $ξ(x)=0$. Regarding $Λ_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(Λ_1)$ arising from it. The code $\mathcal{C}(Λ_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.