Universal optimality of $T$-avoiding spherical codes and designs
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| Format: | Preprint |
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2025
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| author | Boyvalenkov, P. G. Cherkashin, D. D. Dragnev, P. D. |
| author_facet | Boyvalenkov, P. G. Cherkashin, D. D. Dragnev, P. D. |
| contents | Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes--Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of $T$-avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products $α, β, γ$ (in our terminology $(α,β)$-avoiding $γ$-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) $T$-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their $T$-avoiding class for given dimension and minimum distance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13906 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal optimality of $T$-avoiding spherical codes and designs Boyvalenkov, P. G. Cherkashin, D. D. Dragnev, P. D. Combinatorics Information Theory Metric Geometry 05B30 (Primary), 52C17 (secondary) Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes--Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of $T$-avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products $α, β, γ$ (in our terminology $(α,β)$-avoiding $γ$-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) $T$-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their $T$-avoiding class for given dimension and minimum distance. |
| title | Universal optimality of $T$-avoiding spherical codes and designs |
| topic | Combinatorics Information Theory Metric Geometry 05B30 (Primary), 52C17 (secondary) |
| url | https://arxiv.org/abs/2501.13906 |