A New Upper Bound for the Minimum of an Integral Lattice of Determinant One*

Let Λ be an n-dimensional integral lattice of determinant 1. We show that, for all sufficiently large n, the minimal nonzero squared length in Λ does not exceed [ (n + 6 )/10 ]. This bound is a consequence of some new conditions on the theta series of these lattices; these conditions also enable us to find the greatest possible minimal squared length in all dimensions n ≤ 33. In particular we settle the ‘‘no-roots’’ problem: there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n ≥ 23, n ≠ 25. There are also analogues of all these results for codes. ________________ * This paper appeared in Bulletin Amer. Math. Soc., vol. 23 (1990), pp. 383-387, with a correction in vol. 24 (1991), p. 479. A New Upper Bound for the Minimum of an Integral Lattice of Determinant One J. H. Conway Mathematics Department Princeton University Princeton, New Jersey 08540 N. J. A. Sloane Mathematical Sciences Research Center AT&T Bell Laboratories Murray Hill, New Jersey 07974