Since the introduction of the notion of spherical designs by Delsarte, Goethals, and Seidel in 1977, finding explicit constructions of spherical designs had been an open problem. Most existence proofs of spherical designs rely on the topology of the spheres, hence their constructive versions are only computable, but not explicit. That is to say that these constructions can only give algorithms that produce approximations of spherical designs up to arbitrary given precision, while they are not able to give any spherical designs explicitly. Inspired by recent work on rational designs, i.e. designs consisting of rational points, we generalize the known construction of spherical designs that uses interval designs with Gegenbauer weights, and give an explicit formula of spherical designs of arbitrary given strength on the real unit sphere of arbitrary given dimension.

Revised:

Accepted:

Published online:

Keywords: Explicit construction, rational points, spherical designs.

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@article{ALCO_2022__5_2_347_0, author = {Xiang, Ziqing}, title = {Explicit spherical designs}, journal = {Algebraic Combinatorics}, pages = {347--369}, publisher = {The Combinatorics Consortium}, volume = {5}, number = {2}, year = {2022}, doi = {10.5802/alco.213}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.213/} }

TY - JOUR AU - Xiang, Ziqing TI - Explicit spherical designs JO - Algebraic Combinatorics PY - 2022 DA - 2022/// SP - 347 EP - 369 VL - 5 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.213/ UR - https://doi.org/10.5802/alco.213 DO - 10.5802/alco.213 LA - en ID - ALCO_2022__5_2_347_0 ER -

Xiang, Ziqing. Explicit spherical designs. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 347-369. doi : 10.5802/alco.213. https://alco.centre-mersenne.org/articles/10.5802/alco.213/

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