Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding

Let $V$ be a vector space over the finite field ${\mathbb{F}}_q$ with $q$ elements and $\mathrm{\Lambda }$ be the image of the Segre geometry $\mathrm{P}\mathrm{G}(V)\otimes \mathrm{P}\mathrm{G}(V^{\ast })$ in $\mathrm{P}\mathrm{G}(V\otimes V^{\ast })$. Consider the subvariety ${\mathrm{\Lambda }}_1$ of $\mathrm{\Lambda }$ represented by the pure tensors $x\otimes \xi$ with $x\in V$ and $\xi \in V^{\ast }$ such that $\xi (x)=0$. Regarding ${\mathrm{\Lambda }}_1$ as a projective system of $\mathrm{P}\mathrm{G}(V\otimes V^{\ast })$, we study the linear code $\mathcal{C}({\mathrm{\Lambda }}_1)$ arising from it. The code $\mathcal{C}({\mathrm{\Lambda }}_1)$ 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.