Linear codes arising from the point-hyperplane geometry — Part II: the twisted embedding

Let Γ¯ be the point-hyperplane geometry of a projective space PG(V), where V is a (n+1)-dimensional vector space over a finite field Fq of order q . Suppose that σ is an automorphism of Fq and consider the projective embedding εσ of Γ¯ into the projective space PG(V⊗V⁎) mapping the point ([x],[ξ])∈Γ¯ to the projective point represented by the pure tensor xσ⊗ξ, with ξ(x)=0. In [11] , we focused on the case σ=1 and we studied the projective code arising from the projective system Λ1=ε1(Γ¯). Here we focus on the case σ≠1 and we investigate the linear code C(Λσ) arising from the projective system Λσ=εσ(Γ¯). In particular, after having verified that C(Λσ) is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group. We also give a (geometrical) characterization of its minimum and second lowest weight codewords and determine its maximum weight when q and n are both odd.