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

TL;DR

This paper introduces and studies $T$-avoiding spherical codes and designs, proving their universal optimality and minimality in certain classes, with applications to lattices and strongly regular graphs.

Contribution

It defines $T$-avoiding codes and designs, proves their universal optimality, and extends classical results to these restricted classes of spherical codes.

Findings

Certain codes in the Leech and Barnes--Wall lattices are universally optimal within $T$-avoiding classes.

These codes are minimal (tight) $T$-avoiding spherical designs of fixed dimension and strength.

Some codes achieve maximal cardinality in their $T$-avoiding class for given parameters.

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.