二进制子空间啁啾
Binary Subspace Chirps
摘要：
我们详细描述了二进制辛几何和量子计算之间的相互作用，构建高度结构化的码本的最终目标。二进制啁啾（BC）是在无线网络中的确定性压缩感测和随机/无源多址中使用的$N = 2^m$维度上的复数格拉斯曼线。它们的项是单位的四阶根，并且可以用二阶Reed-Muller码来描述。二进制子空间线性调频（BSSC）是范围从$r=0$到$r = m$的$\textit{ranks}$的BC的唯一集合，根据由秩$r$二进制子空间确定的开关模式嵌入在$N$维度中。这产生的码本渐进地比BC的码本大2.38倍，具有与BC的码本相同的最小弦距离，并且字母表从$\{\pm 1,\pm i\}$最低限度地扩展到$\{\pm 1,\pm i, 0\}$。等价地，我们表明，BSSC的稳定状态，我们表征它们作为列的一个良好的控制收集克利福德矩阵。通过构造，BSSC继承了BC的所有属性，这反过来使它们成为各种应用的良好候选者。对于无线通信中的应用，我们利用BSSC丰富的代数结构来构造一个低复杂度的解码算法，该算法对高斯噪声是可靠的。在模拟中，BSSC表现出与BC相当或略低于BC的错误概率，无论是对于单用户还是多用户传输。
英文摘要：
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.
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