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PreprintCC BY 4.0arXiv

A Note on Sphere Packing Bounds for Tuple Lattice Sieving

  • Thijs Laarhoven
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Abstract

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 ℛk 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: ℛk ≤ min1 ≤ r ≤ ⌊ k/2 ⌋ 1/r κ (1 − 1/2r) . Combining this with standard sphere packing bounds, for large k we obtain an almost-tight asymptotic comparison with the known lower bounds: (1/2−o(1)) (log2 k)/k ≤ ℛk ≤ (1 + o(1)) (log2 k)/k .

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Cite

BibTeXRIS
BibTeX
@misc{laarhoven2026note,
  title = {{A Note on Sphere Packing Bounds for Tuple Lattice Sieving}},
  author = {Thijs Laarhoven},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08190},
  eprint = {2609.08190},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08190},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Note on Sphere Packing Bounds for Tuple Lattice Sieving
AU  - Thijs Laarhoven
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08190
UR  - https://arxiv.org/abs/2609.08190
AB  - 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 ≤ min_1 ≤ r ≤ ⌊ k/2 ⌋ 1/r κ (1 − 1/2r) . Combining this with standard sphere packing bounds, for large k we obtain an almost-tight asymptotic comparison with the known lower bounds: (1/2−o(1)) (log_2 k)/k ≤ ℛ_k ≤ (1 + o(1)) (log_2 k)/k .
ER  -