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We show that, if L is an extremal Type II lattice of rank 40 or 80, then L isgenerated by its vectors of norm min(L)+2. This sharpens earlier results ofOzeki, and the second author and Abel, which showed that such lattices L aregenerated by their vectors of norms min(L) and min(L)+2.

Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80