Hadamard Equiangular Tight Frames

An equiangular tight frame (ETF) is a type of optimal packing of lines. 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 unimodular entries. 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. In this paper, we give some new results about flat ETFs. We give 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. Published by Elsevier Inc.