Document Type
Article - preprint
Department
Mathematics (Pomona), Mathematics (CMC)
Publication Date
2016
Keywords
Lattice, Equiangular lines, Tight frame, Conference matrix
Abstract
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.
Rights Information
© 2016 Elsevier Inc.
DOI
10.1016/j.laa.2016.09.008
Recommended Citation
Böttcher, A., Fukshansky, L., Garcia, S.R., Maharaj, H., Needell, D., Lattices from tight equiangular frames, Lin. Alg. Appl. 510 (2016), 395-420.
Comments
Final published version can be found at: Böttcher, A., Fukshansky, L., Garcia, S.R., Maharaj, H., Needell, D., Lattices from tight equiangular frames, Lin. Alg. Appl. 510 (2016), 395-420.