PreprintCC BY 4.0Zenodo
Spherical codes with minimal angle arccos(1/3) in dimensions four, five and six
Abstract
Let A(n, θ) be the largest number of points on the unit sphere in R^n with pairwise angular distance at least θ. We prove A(4, arccos 1/3) = 14, A(5, arccos 1/3) ≤ 22 and A(6, arccos 1/3) ≤ 34. The previous upper bounds were 15, 23 and 35. For n = 5 the best known code has 21 points, so A(5, arccos 1/3) is 21 or 22. Each upper bound comes from an exact rational solution of the second level of the Lasserre hierarchy for spherical codes, in the formulation of de Laat, Leijenhorst and de Muinck Keizer, and was checked with their verification program. The largest number of points at angle π/3 from a given point of a kissing configuration in R^n equals A(n-1, arccos 1/3). So this number is 14 in R^5, 21 or 22 in R^6, and at most 34 in R^7.
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BibTeX
@misc{tomar2026spherical,
title = {{Spherical codes with minimal angle arccos(1/3) in dimensions four, five and six}},
author = {Tejasvi Singh Tomar},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23603435},
url = {https://zenodo.org/records/23603435},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spherical codes with minimal angle arccos(1/3) in dimensions four, five and six
AU - Tejasvi Singh Tomar
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23603435
UR - https://zenodo.org/records/23603435
AB - Let A(n, θ) be the largest number of points on the unit sphere in R^n with pairwise angular distance at least θ. We prove A(4, arccos 1/3) = 14, A(5, arccos 1/3) ≤ 22 and A(6, arccos 1/3) ≤ 34. The previous upper bounds were 15, 23 and 35. For n = 5 the best known code has 21 points, so A(5, arccos 1/3) is 21 or 22. Each upper bound comes from an exact rational solution of the second level of the Lasserre hierarchy for spherical codes, in the formulation of de Laat, Leijenhorst and de Muinck Keizer, and was checked with their verification program. The largest number of points at angle π/3 from a given point of a kissing configuration in R^n equals A(n-1, arccos 1/3). So this number is 14 in R^5, 21 or 22 in R^6, and at most 34 in R^7.
ER -