  An equiangular tight frame (ETF) is a set of unit vectors in a Euclidean
space whose coherence is as small as possible, equaling the Welch bound. Also
known as Welch-bound-equality sequences, such frames arise in various
applications, such as waveform design and compressed sensing. At the moment,
there are only two known flexible methods for constructing ETFs: harmonic ETFs
are formed by carefully extracting rows from a discrete Fourier transform;
Steiner ETFs arise from a tensor-like combination of a combinatorial design and
a regular simplex. These two classes seem very different: the vectors in
harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely
sparse. We show that they are actually intimately connected: a large class of
Steiner ETFs can be unitarily transformed into constant-amplitude frames,
dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs
is a subset of an important class of Kirkman ETFs. This connection informs the
discussion of both types of frames: some Steiner ETFs can be transformed into
constant-amplitude waveforms making them more useful in waveform design; some
harmonic ETFs have low spark, making them less desirable for compressed
sensing. We conclude by showing that real-valued constant-amplitude ETFs are
equivalent to binary codes that achieve the Grey-Rankin bound, and then
construct such codes using Kirkman ETFs.
