关于元组格筛法的球堆积界的一个注记
A Note on Sphere Packing Bounds for Tuple Lattice Sieving
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中文总结 AI 辅助
本文针对元组格筛法中的球堆积问题，定义了$k$-不可约向量集，证明了其最大速率$\mathcal{R}_k$的上界，并结合标准球堆积界，给出了与已知下界几乎紧的渐近比较。
中文摘要 AI 辅助
一个有限单位向量集合称为$k$-不可约的，如果任意介于2到$k$个不同元素的带符号和之范数大于1。设$\mathcal{R}_k$为这类集合的最大渐近速率，并设$\kappa(\alpha)$为两两内积至多为$\alpha$的球码的最大渐近速率。对于$k \ge 2$，我们证明：\begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \\, \kappa\\!\left(1 - \frac{1}{2r}\right) \\,. \end{align} 将此与标准球堆积界结合，对于大的$k$，我们获得与已知下界几乎紧的渐近比较：\begin{align} \left(\tfrac{1}{2}-o(1)\right) \\, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \\, \frac{\log_2 k}{k} \\,. \end{align}
英文摘要
A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $κ(α)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $α$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, κ\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}
