Lattices from equiangular tight frames

We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight (k,n)(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+1n=k+1 and that there are infinitely many k   such that a lattice emerges for n=2kn=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.