1999/06/12 by Noam D. Elkies
Mathematics · #math.NT #msc:11H55 #msc:11F11 #msc:11H06 #msc:94B05
published as Math. Research Letters 2 (1995), 643-651 · 8 pages. Note: Mark Gaulter has since established the existence of integers N_k also for k=2,3
arxiv created 1999/06/12 · arxiv updated 2009/11/30
In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Zn has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".