Binary Subspace Chirps

Tefjol Pllaha, Olav Tirkkonen, Robert Calderbank

Research output: Contribution to journalArticleScientificpeer-review

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 N = 2m 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 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 {±1,±i} to {±1,±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.

Original languageEnglish
JournalIEEE Transactions on Information Theory
DOIs
Publication statusE-pub ahead of print - 2022
MoE publication typeA1 Journal article-refereed

Keywords

  • Chirp
  • Complexity theory
  • Decoding
  • Geometry
  • Interference
  • Symmetric matrices
  • Transforms

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