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In this paper we consider the extremal even self-dual ${𝔽}_4$-additive codes. We give a complete classification for length $10$. Under the hypothesis that at least two minimal words have the same support, we classify the codes of length $14$ and we show that in length $18$ such a code is equivalent to the unique ${𝔽}_4$-hermitian code with parameters [18,9,8]. We construct with the help of them some extremal $3$-modular lattices.

We consider the problem of constructing dense lattices in ${ℝ}^n$ with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least $cn{2}^{-n}$, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions $n$, we exhibit a finite set of lattices that come with an automorphisms group of size $n$, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic...

The alphabet ${𝐅}_3+v{𝐅}_3$ where $v^2=1$ is viewed here as a quotient of the ring of integers of $𝐐\left(\sqrt{-2}\right)$ by the ideal (3). Self-dual ${𝐅}_3+v{𝐅}_3$ 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...