Lattices and codes with long shadows

Abstract

In an earlier paper we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n 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".