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)⊗PG(V⁎) in PG(V⊗V⁎) under the Segre map. 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. We show that C(Λ1) is a minimal code and we determine its basic parameters, its full 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.