Binary Subspace Chirps
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Abstract
We describe in detail the interplay between binary symplectic geometry and notions from quantum computation, with the ultimate goal of constructing highly structured codebooks. The Binary Chirps (BCs) are Complex Grassmannian Lines in <inline-formula> <tex-math notation="LaTeX">$N = 2^{m}$ </tex-math></inline-formula> 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 ranks ranging from <inline-formula> <tex-math notation="LaTeX">$r=0$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$r = m$ </tex-math></inline-formula>, embedded in <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> dimensions according to an on-off pattern determined by a rank <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\{\pm 1,\pm i\}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$\{\pm 1,\pm i, 0\}$ </tex-math></inline-formula>. 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.