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

Abstract

Let $V$ be a vector space over the finite field ${\mathbb{F}}_q$ with $q$ elements and $\hat{I}›$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes \mathrm{PG}(V^{\ast })$ in $\mathrm{PG}(V\otimes V^{\ast })$. Consider the subvariety $\hat{I}{›}_1$ of $\hat{I}›$ represented by the pure tensors $x\otimes \hat{I}¾$ with $x\in V$ and $\hat{I}¾\in V^{\ast }$ such that $\hat{I}¾(x)=0$. Regarding $\hat{I}{›}_1$ as a projective system of $\mathrm{PG}(V\otimes V^{\ast })$, we study the linear code $\mathcal{C}(\hat{I}{›}_1)$ arising from it. The code $\mathcal{C}(\hat{I}{›}_1)$ 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.

31 pages/revised version