Odd Systems of Vectors and Related Lattices

AbstractWe consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(V) linearly generated by a uniform odd system V of odd norm 2t+1. If uu ≡ p (mod 4) for all u ∈ V, one has v2 ≡ p (mod 4) if v2 is odd and v2 ≡ 0 (mod 4) if v2 is even, for any vector v ∈ L(V). The vectors of even norm form a double even sublattice L0(V) of L(V), i.e. $$(1/\sqrt 2 )L_0 (\mathcal{V}) $$ is an even lattice. The closure of V, i.e. all vectors of L(V) of norm 2t+1, are minimal vectors of L(V) for t=1, and they are almost always minimal for t=2. For such t, the convex hull of vectors of the closure of V is an L-polytope of L0V and the contact polytope of L(V). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.