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arXiv · 1607.05350

Lattices from tight equiangular frames

Albrecht Boettcher·Lenny Fukshansky·Stephan Ramon Garcia·Hiren Maharaj·Deanna Needell
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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.

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Albrecht Boettcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell. 2016-07-18. Lattices from tight equiangular frames. https://arxiv.org/abs/1607.05350

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