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arXiv · 2501.13906

Universal optimality of $T$-avoiding spherical codes and designs

P. G. Boyvalenkov·D. D. Cherkashin·P. D. Dragnev
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Abstract

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 $\alpha, \beta, \gamma$ (in our terminology $(\alpha,\beta)$-avoiding $\gamma$-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.

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BibTeXRIS

P. G. Boyvalenkov, D. D. Cherkashin, P. D. Dragnev. 2025-01-23. Universal optimality of $T$-avoiding spherical codes and designs. https://arxiv.org/abs/2501.13906

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