Self-orthogonal flags of codes and translation of flags of algebraic geometry codes

A flag C₀ C₁ ⋯ Cₛ Fqⁿ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy Cᵢ^⊥=Cₛ₋ᵢ, and it is said to satisfy the isometry-dual property with respect to an isometry vector x if Cᵢ^⊥= x Cₛ₋ᵢ for i=1, , s. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of Fqⁿ. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.