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.

等角紧密框架（ETF）是一种最佳的线形堆积。它们通常表示为简短的胖矩阵的列。在某些应用中，我们希望此矩阵是平坦的，即具有单模项。特别是，真正的扁平ETF等效于达到Grey-Rankin界的自互补二进制代码。一些扁平ETF是（复杂）Hadamard ETF，这意味着它们是通过从（复杂）Hadamard矩阵中提取行而产生的。在本文中，我们给出了有关扁平ETF的一些新结果。我们为所有Steiner ETF提供了明确的Naimark补充，这反过来意味着所有Kirkman ETF可能都是复杂的Hadamard ETF。这尤其产生了一个新的无限系列的真实扁平ETF。另一个结果确定了真正的扁平ETF与某些类型的准对称设计之间的等效性，