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.

等角紧框架 (ETF) 是希尔伯特空间中的向量序列，它在韦尔奇界中实现相等，因此具有最小的相干性。更一般地，等弦紧融合框架 (ECTFF) 是 Hilbert 空间的等维子空间序列，它在 Conway、Hardin 和 Sloane 的单纯形界中实现相等。每个 ECTFF 都是一种最优格拉斯曼码，即希尔伯特空间的等维子空间的最优打包。我们通过利用已知 ETF 之间的新关系构建 ECTFF。谐波 ETF 等同于有限阿贝尔群的差分集。当其谐波 ETF 的相应子序列恰好是其跨度的 ETF 时，我们说这种组的差集与其 Pontryagin 对偶的差集“配对”。我们表明，每个这样的对都会产生一个 ECTFF。此外，我们在两个元素的域上使用二次形式构造了一个无限的成对差分集族。这一起产生了两个无限的真实 ECTFF 系列。