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 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 2 we show: R k 1 r k/2 1 over r, κ! (1 - 1 over 2r),. Combining this with standard sphere packing bounds, for large k we obtain an almost-tight asymptotic comparison with the known lower bounds: (2-o(1)), 2 k over k R k (1 + o(1)), 2 k over k,.