  An equiangular tight frame (ETF) is a type of optimal packing of lines in a
finite-dimensional Hilbert space. ETFs arise in various applications, such as
waveform design for wireless communication, compressed sensing, quantum
information theory and algebraic coding theory. In a recent paper, signature
matrices of ETFs were constructed from abelian distance regular covers of
complete graphs. We extend this work, constructing ETF synthesis operators from
abelian generalized quadrangles, and vice versa. This produces a new infinite
family of complex ETFs as well as a new proof of the existence of certain
generalized quadrangles. This work involves designing matrices whose entries
are polynomials over a finite abelian group. As such, it is related to the
concept of a polyphase matrix of a finite filter bank.
