Combinatorics of Delaunay polytopes of the isodual lattice Q 10

Abstract

The results of [9] are generalized and simplified for code lattices. As an example, the code lattice Q 10 , mentioned and named in the paper [6], is considered. Q 10 has two symmetric Delaunay polytopes P 5 , P 3 and an asymmetric P 0 5 , and is generated by P 5 . P 5 is a symmetrization of the cut polytope PCut 5 , i.e. it is the convex hull of all cuts and their complements in the complete graph K 5 . The cuts and their complements are all circuits of the regular matroid R 10 [12]. Besides P 5 is the convex hull of the unique 10-dimensional closed odd system of 16 pairs of opposite vectors of norm 5 spanning equiangular lines at angle arccos 1 5 . P 5 is also the convex hull of all codewords of the linear binary code C 10 = [10; 5; 4]. The second symmetric Delaunay polytopes P 3 is the convex hull of a system of 40 pairs of opposite vectors of norm 3 with 0; \Sigma1 inner products. The asymmetric Delaunay polytope P 0 5 is the convex hull of an odd system of 32 vectors of norm ..