Binary Subspace Chirps

TL;DR

This paper introduces Binary Subspace Chirps, a new class of structured codebooks derived from binary symplectic geometry, with applications in wireless communication and compressed sensing, offering larger size and reliable decoding.

Contribution

The paper presents Binary Subspace Chirps, extending Binary Chirps with higher ranks, increasing codebook size, and characterizing them as stabilizer states with efficient decoding algorithms.

Findings

BSSCs are asymptotically 2.38 times larger than BCs.

BSSCs maintain the same minimum chordal distance as BCs.

Simulations show BSSCs have comparable or lower error probability than BCs.

Abstract

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.