  An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space
that achieves equality in the Welch bound and so has minimal coherence. More
generally, an equichordal tight fusion frame (ECTFF) is a sequence of
equi-dimensional subspaces of a Hilbert space that achieves equality in Conway,
Hardin and Sloane's simplex bound. Every ECTFF is a type of optimal
Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of
a Hilbert space. We construct ECTFFs by exploiting new relationships between
known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups.
We say that a difference set for such a group is "paired" with a difference set
for its Pontryagin dual when the corresponding subsequence of its harmonic ETF
happens to be an ETF for its span. We show that every such pair yields an
ECTFF. We moreover construct an infinite family of paired difference sets using
quadratic forms over the field of two elements. Together this yields two
infinite families of real ECTFFs.
