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 $\Lambda$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $\Lambda_{1}$ of $\Lambda$ represented by the pure tensors $x\otimes \xi$ with $x\in V$ and $\xi\in V^*$ such that $\xi(x)=0$. Regarding $\Lambda_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(\Lambda_1)$ arising from it. The code $\mathcal{C}(\Lambda_1)$ 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.