A Note on Sphere Packing Bounds for Tuple Lattice Sieving

RESUMO

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 $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha$. For $k \ge 2$ we show: \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} 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}