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 F q of order q . Suppose that σ is an automorphism of F q and consider the projective embedding ε σ of Γ ¯ into the projective space PG ( V ⊗ V ⁎ ) mapping the point ( [ x ] , [ ξ ] ) ∈ Γ ¯ to the projective point represented by the pure tensor x σ ⊗ 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 Λ 1 = ε 1 ( Γ ¯ ) . Here we focus on the case σ ≠ 1 and we investigate the linear code C ( Λ σ ) arising from the projective system Λ σ = ε σ ( Γ ¯ ) . In particular, after having verified that C ( Λ σ ) 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.