Lattices without short characteristic vectors

All the lattices here under discussion here are understood to be integral unimodular $\Bbb Z$-lattices in ${\Bbb R} ^n$. A {\it characteristic vector} of a lattice $L$ is a vector $w \in L$ such that $v\cdot w\equiv |v|^2 \pmod 2$ for every $v\in L$. Elkies has considered the minimal (squared) norm of the characteristic vectors in a unimodular lattice. He showed that any unimodular ${\Bbb Z}$-lattice in ${\Bbb R} ^n$ has characteristic vectors of norm $\le n$; he also proved that of all such lattices, only the standard lattice ${\Bbb Z}^n$ has no characteristic vectors of norm $<n$ ({\it Math Research Letters} {\bf 2}, 321-326). He then asked “For any $k >0$, is there ${\cal N} _k$ such that every integral unimodular lattice all of whose characteristic vectors have norm $\ge n-8k$ is of the form $L_0 \perp {\Bbb Z}^r$ for some lattice $L_0$ of rank at most ${\cal N}_k$?” ({\it Math Research Letters} {\bf 2}, 643-651). He solved this question in the case $k=1$, showing that ${\cal N}_1 =23$ suffices; here I determine values for ${\cal N}_2$ and ${\cal N}_3$.

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Published 1 January 1998