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Even lattices and doubly even codes. (English) Zbl 0726.94009
In this paper, we study three methods to construct even lattices in an n- dimensional Euclidean space $$E^ n$$ from doubly even codes of length n. The first two methods are the constructions A, B appeared in Conway- Sloane’s book “Sphere packings, lattices and groups”, and the third method is a generalization of the construction of the Leech lattice from the Golay code.
We prove that non-equivalent codes yield non-equivalent lattices except some small dimension cases. Some counter examples are given. We consider the structure of the code, and construct some automorphisms of the lattice. Our main results follow from the transitivity of the automorphism group on some family of orthogonal bases of the Euclidean space $$E^ n$$.
Reviewer: Masaaki Kitazume

##### MSC:
 94B05 Linear codes, general 94B15 Cyclic codes 51M05 Euclidean geometries (general) and generalizations
##### Keywords:
even lattices; doubly even codes; Leech lattice; Golay code
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