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A note on optimal unimodular lattices. (English) Zbl 0917.11027

The authors provide explicit relations between the theta series of a unimodular lattice and that of its shadow. A careful examination of the coefficients involved leads to upper bounds on the highest possible minimal norm of a unimodular lattice in a given dimension \(n\) (here the authors are concerned with \(n\leq 40\)). Unimodular lattices achieving this bound, i.e. optimal unimodular lattices, in dimensions \(\leq 40\) are then discussed. It is also noted that, to date, it is not known whether there exist optimal unimodular lattices in dimensions 34, 35, 37, 38, or 39.
The results of this paper were announced in [J. H. Conway and N. J. A. Sloane, Bull. Am. Math. Soc. 23, 383-387 (1990; Zbl 0709.11030); Erratum, ibid. 24, 479 (1991; Zbl 0719.11021)].
Reviewer: K.Roegner (Berlin)

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E25 Sums of squares and representations by other particular quadratic forms
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