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

P. G. Boyvalenkov, D. D. Cherkashin, P. D. Dragnev

TL;DR

This work introduces and studies $T$-avoiding spherical codes and designs, proving their universal optimality within restricted inner-product sets for a variety of highly symmetric constructions. It develops a general linear programming framework using Gegenbauer polynomials to bound maximal cardinality, minimal tight designs, and minimal energy, and applies it to codes derived from the Leech lattice, Barnes–Wall lattice, and strongly regular graphs. The paper identifies concrete, tight $T$-avoiding designs in dimensions $22$, $21$, and $15$, and proves universal optimality of multiple codes (including derived Leech codes, Golay complement, BW, and SRG embeddings) for specific $T$-intervals and absolutely monotone potentials. These results unify extremal properties (maximal size, tight designs, and minimal energy) within a single LP-analytic framework and provide sharp bounds and equalities for a suite of classical and newly constructed spherical codes. The findings have implications for kissing configurations, design theory, and energy minimization in highly symmetric spherical configurations.

Abstract

Given an open set (a union of open intervals), $T\subset [-1,1]$ 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 lattices, 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.

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

TL;DR

This work introduces and studies -avoiding spherical codes and designs, proving their universal optimality within restricted inner-product sets for a variety of highly symmetric constructions. It develops a general linear programming framework using Gegenbauer polynomials to bound maximal cardinality, minimal tight designs, and minimal energy, and applies it to codes derived from the Leech lattice, Barnes–Wall lattice, and strongly regular graphs. The paper identifies concrete, tight -avoiding designs in dimensions , , and , and proves universal optimality of multiple codes (including derived Leech codes, Golay complement, BW, and SRG embeddings) for specific -intervals and absolutely monotone potentials. These results unify extremal properties (maximal size, tight designs, and minimal energy) within a single LP-analytic framework and provide sharp bounds and equalities for a suite of classical and newly constructed spherical codes. The findings have implications for kissing configurations, design theory, and energy minimization in highly symmetric spherical configurations.

Abstract

Given an open set (a union of open intervals), we introduce the concepts of -avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set . We show that certain codes found in the minimal vectors of the Leech lattices, 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 -avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products (in our terminology -avoiding -codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) -avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their -avoiding class for given dimension and minimum distance.
Paper Structure (26 sections, 34 theorems, 212 equations)

This paper contains 26 sections, 34 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General linear programming theorems
  4. Maximal cardinality of T-avoiding codes
  5. Minimum cardinality of T-avoiding designs
  6. Minimum energy of T-avoiding codes
  7. Examples in 48 dimensions
  8. Codes
  9. Codes on S22
  10. On the orthogonal complement of the Golay code
  11. Codes on S21
  12. Codes on S15
  13. Codes from strongly regular graphs
  14. Maximal T-avoiding codes
  15. Codes on S22
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $C$ be an $(n, N, d, \tau)$-configuration with $\tau \geq d - 1$. Then $C$ is distance invariant.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1: part of Theorem 7.4 in DGS
  • Theorem 2.2: Theorem 8.2 in DGS
  • Theorem 2.3: Theorem 3.2 in BBDHSS2025
  • Theorem 3.1: LP for unrestricted codes DGS
  • Theorem 3.2: LP for spherical designs DGS
  • Definition 3.3
  • ...and 40 more