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Unimodular lattices with a complete root system. (English) Zbl 0808.11024
Let $$\Lambda$$ be a lattice in $$\mathbb{R}^ n$$ with root system $$\Lambda_ 2$$. Then $$\Lambda_ 2$$ is called complete if $$\Lambda_ 2$$ generates $$\mathbb{R}^ n$$ as an $$\mathbb{R}$$-vector space. The author gives a classification of even unimodular lattices of dimension 32 with complete root system. If $$\Lambda_ 2$$ contains an orthogonal basis of $$\mathbb{R}^ n$$, then the classification of lattices is equivalent to the classification of doubly-even self-dual binary codes of length 32, which was accomplished by J. H. Conway and V. Pless [J. Comb. Theory, Ser. A 28, 26-53 (1980; Zbl 0439.94011)]. The problem of classifying lattices without orthogonal bases remains. There are exactly 57 such lattices which are indecomposable. They are all determined by their root system.
As in the classification of codes in the above cited paper of Conway and Pless the author needed the help of a computer to get his result.
Reviewer: H.Koch (Berlin)

##### MSC:
 11E12 Quadratic forms over global rings and fields 94B40 Arithmetic codes