# Lattices and codes with long shadows

 Title: Lattices and codes with long shadows Author: Elkies, Noam Citation: Elkies, Noam D. 1995. Lattices and codes with long shadows. Mathematical Research Letters 2(5): 643-651. Full Text & Related Files: Elkies - Lattices and codes.pdf (138.3Kb; PDF) 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". Other Sources: http://www.mrlonline.org/mrl/1995-002-005/index.html http://arxiv.org/abs/math/9906086 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2961697 Downloads of this work:

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