Lattices and quadratic forms from tight frames in Euclidean spaces

TL;DR

This paper explores the relationship between tight frames in Euclidean spaces and the resulting lattices, establishing conditions for when these lattices are eutactic, perfect, and local maxima of packing density, with specific results in low dimensions.

Contribution

It proves that the integral span of an equiangular tight frame is a lattice if and only if the frame is rational, and analyzes conditions for lattices to be eutactic and perfect.

Findings

Integral span of ETF is a lattice iff the frame is rational.

The (276, 23) ETF is eutactic and perfect.

Results obtained for tight frames in dimensions two and three.

Abstract

This paper supplies additions to our paper in Linear Algebra Appl. 510 (2016) 395--420 on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit (276, 23) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three.