A Note on Sphere Packing Bounds for Tuple Lattice Sieving

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} , 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}