Odd Systems of Vectors and Related Lattices

摘要

We consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(ν) linearly generated by a uniform odd system ν of odd norm st + 1. If u~2 ≡ p (mod 4) for all u ∈ ν, one has v~2 ≡ p (mod 4) if v~2 is odd and v~2 ≡ 0 (mod 4) if v~2 is even, for any vector v ∈ L(ν). The vectors of even norm form a double even sublattice L_0(ν) of L(ν), i.e. (1/2~(1/2))L_0(ν) is an even lattice. The closure of ν, i.e. all vectors of L(ν) of norm 2t + 1, are minimal vectors of L(ν) for t = 1, and they are almost always minimal for t = 2. For such t, the convex hull of vectors of the closure of ν is a L-polytope of L_0(ν) and the contact polytope of L(ν). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.