  An equiangular tight frame (ETF) is a type of optimal packing of lines in
Euclidean space. They are often represented as the columns of a short, fat
matrix. In certain applications we want this matrix to be flat, that is, have
the property that all of its entries have modulus one. In particular, real flat
ETFs are equivalent to self-complementary binary codes that achieve the
Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they
arise by extracting rows from a (complex) Hadamard matrix. These include
harmonic ETFs, which are obtained by extracting the rows of a character table
that correspond to a difference set in the underlying finite abelian group. In
this paper, we give some new results about flat ETFs. One of these results
gives an explicit Naimark complement for all Steiner ETFs, which in turn
implies that all Kirkman ETFs are possibly-complex Hadamard ETFs. This in
particular produces a new infinite family of real flat ETFs. Another result
establishes an equivalence between real flat ETFs and certain types of
quasi-symmetric designs, resulting in a new infinite family of such designs.
