Article - preprint
Mathematics (Pomona), Mathematics (CMC)
Lattice, Equiangular lines, Tight frame, Conference matrix
We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k,n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.
© 2016 Elsevier Inc.
Böttcher, A., Fukshansky, L., Garcia, S.R., Maharaj, H., Needell, D., Lattices from tight equiangular frames, Lin. Alg. Appl. 510 (2016), 395-420.