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

Let V be a vector space over the finite field Fq with q elements and $Λ$ be the image of the Segre geometry PG(V)(V^*) in PG(V⊗ V^*). Consider the subvariety Λ₁ of $Λ$ represented by the pure tensors x⊗ ξ with x∈ V and ξ∈ V^* such that $ξ(x)=0$. Regarding Λ₁ as a projective system of PG(V⊗ V^*), we study the linear code C(Λ₁) arising from it. The code C(Λ₁) 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.