### $2$-modular lattices from ternary codes

The alphabet ${\mathbf{F}}_{3}+v{\mathbf{F}}_{3}$ where ${v}^{2}=1$ is viewed here as a quotient of the ring of integers of $\mathbf{Q}\left(\sqrt{-2}\right)$ by the ideal (3). Self-dual ${\mathbf{F}}_{3}+v{\mathbf{F}}_{3}$ codes for the hermitian scalar product give $2$-modular lattices by construction ${A}_{K}$. There is a Gray map which maps self-dual codes for the Euclidean scalar product into Type III codes with a fixed point free involution in their automorphism group. Gleason type theorems for the symmetrized weight enumerators of Euclidean self-dual codes and the length weight enumerator of hermitian self-dual...