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.

中文翻译：

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二进制子空间Chi

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，最终目的是构建高度结构化的代码本。二进制线性调频（BCs）是维度为N = 2 ^ m $的复草曼线，用于确定性压缩感知和无线网络中的随机/无源多址访问。它们的输入是统一的第四根，可以用二阶里德-穆勒码来描述。Binary Subspace Chirps（BSSCs）是$ \ textit {ranks $}的BC的唯一集合，范围从$ r = 0 $到$ r = m $，并根据由-n决定的开关模式嵌入到$ N $维度中。等级$ r $二进制子空间。这样产生的码本渐近是BCs码本的2.38倍，并具有与BCs码本相同的最小弦长，并且字母从$ \ {\ pm 1，\ pm i \} $最小扩展到$ \ {\ pm 1，\ pm i，0 \} $。等效地，我们表明BSSC是稳定器状态，并将它们表征为Clifford矩阵的一个受控良好集合的列。通过构造，BSSC继承了BC的所有属性，从而使它们成为各种应用程序的理想候选者。对于无线通信中的应用，我们使用BSSC的丰富代数结构来构建对高斯噪声可靠的低复杂度解码算法。在仿真中，对于单用户和多用户传输，BSSC的错误概率均与BC相当或略低。我们将它们描述为控制良好的Clifford矩阵集合的列。通过构造，BSSC继承了BC的所有属性，从而使它们成为各种应用程序的理想候选者。对于无线通信中的应用，我们使用BSSC的丰富代数结构来构建对高斯噪声可靠的低复杂度解码算法。在仿真中，对于单用户和多用户传输，BSSC的错误概率均与BC相当或略低。我们将它们描述为控制良好的Clifford矩阵集合的列。通过构造，BSSC继承了BC的所有属性，从而使它们成为各种应用程序的理想候选者。对于无线通信中的应用，我们使用BSSC的丰富代数结构来构建对高斯噪声可靠的低复杂度解码算法。在仿真中，对于单用户和多用户传输，BSSC的错误概率均与BC相当或略低。我们使用BSSC的丰富代数结构来构建可抵抗高斯噪声的低复杂度解码算法。在仿真中，对于单用户和多用户传输，BSSC的错误概率均与BC相当或略低。我们使用BSSC的丰富代数结构来构建可抵抗高斯噪声的低复杂度解码算法。在仿真中，对于单用户和多用户传输，BSSC的错误概率均与BC相当或略低。