Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs
Abstract
We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension (2m+1)2−2. Let m be a positive even integer satisfying ν2(m(m+1))≤5, where νp(a) denotes the exponent of the prime p in the positive integer a. We show that such a design does not exist if either (i) m≡1pmod3 and νp(m(m+1))≤1 for every prime p≥7 with p≡3pmod4, or (ii) m≡0pmod3, ν3(m(m+1))=1, and νp(m(m+1))≤1 for every prime p≥5 with pnot≡1pmod12. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including m=16,40,88,96,100. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.
Related
- Seven ordered points and the nonexistence of arithmetic ball quotient compactifications
- Weighted configurations and the nonexistence of arithmetic ball quotient compactifications
- Nonexistence of strongly regular graphs via multipoint spherical semidefinite bounds
- Classification of tight 2 s 2s 2 s-designs with s ≥ 2 s ≥ 2 s greater than or equals 2
- Efficient lifts into odd dimensions of Spherical $5$- and $7$-designs with Balanced Halves
BibTeX
@misc{xu2026arithmetic,
title = {{Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs}},
author = {Zili Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36725},
eprint = {2609.36725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36725},
note = {CC0},
}
RIS
TY - UNPB
TI - Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs
AU - Zili Xu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.36725
UR - https://arxiv.org/abs/2609.36725
AB - We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension (2m+1)^2−2. Let m be a positive even integer satisfying ν_2(m(m+1))≤5, where ν_p(a) denotes the exponent of the prime p in the positive integer a. We show that such a design does not exist if either (i) m≡1pmod3 and ν_p(m(m+1))≤1 for every prime p≥7 with p≡3pmod4, or (ii) m≡0pmod3, ν_3(m(m+1))=1, and ν_p(m(m+1))≤1 for every prime p≥5 with pnot≡1pmod12. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including m=16,40,88,96,100. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.
ER -