  We describe in details the interplay between binary symplectic geometry and
quantum computation, with the ultimate goal of constructing highly structured
codebooks. The Binary Chirps (BCs) are Complex Grassmannian Lines in $N = 2^m$
dimensions used in deterministic compressed sensing and random/unsourced
multiple access in wireless networks. Their entries are fourth roots of unity
and can be described in terms of second order Reed-Muller codes. The Binary
Subspace Chirps (BSSCs) are a unique collection of BCs of $\textit{ranks}$
ranging from $r=0$ to $r = m$, embedded in $N$ dimensions according to an
on-off pattern determined by a rank $r$ binary subspace. This yields a codebook
that is asymptotically 2.38 times larger than the codebook of BCs, has the same
minimum chordal distance as the codebook of BCs, and the alphabet is minimally
extended from $\{\pm 1,\pm i\}$ to $\{\pm 1,\pm i, 0\}$. Equivalently, we show
that BSSCs are stabilizer states, and we characterize them as columns of a
well-controlled collection of Clifford matrices. By construction, the BSSCs
inherit all the properties of BCs, which in turn makes them good candidates for
a variety of applications. For applications in wireless communication, we use
the rich algebraic structure of BSSCs to construct a low complexity decoding
algorithm that is reliable against Gaussian noise. In simulations, BSSCs
exhibit an error probability comparable or slightly lower than BCs, both for
single-user and multi-user transmissions.
