Codebooks of Complex Lines Based on Binary Subspace Chirps

Motivated by problems in machine-type wireless communications, we consider codebooks of complex Grassmannian lines in $N = 2^{m}$ dimensions. Binary Chirp (BC) codebooks of prior art are expanded to codebooks of Binary Subspace Chirps (BSSCs), where there is a binary chirp in a subset of the dimensions, while in the remaining dimensions there is a zero. BSSC codebooks have the same minimum distance as BC codebooks, while the cardinality is asymptotically 2.38 times larger. We discuss how BC codebooks can be understood in terms of a subset of the binary symplectic group $\mathrm{S}\mathrm{p}(2m,\ 2)$ in 2m dimensions; $\mathrm{S}\mathrm{p}(2m,\ 2)$ is isomorphic to a quotient group of the Clifford group acting on the codewords in N dimensions. The Bruhat decomposition of $\mathrm{S}\mathrm{p}(2m,\ 2)$ can be described in terms of binary subspaces in m dimensions, with ranks ranging from $r=0$ to $r=m$. We provide a unique parameterization of the decomposition. The BCs arise directly from the full-rank part of the decomposition, while BSSCs are a group code arising from the action of the full group with generic r. The rank of the binary subspace is directly related to the number of zeros (sparsity) in the BSSC. We develop a reconstruction algorithm that finds the correct codeword with $O(N\log^{2}N)$ complexity, and present performance results in an additive white Gaussian noise scenario.