On the strong non-rigidity of certain tight Euclidean designs

TL;DR

This paper investigates the non-rigidity of certain tight Euclidean t-designs, revealing their ability to be deformed while maintaining their design properties, and provides a complete classification of tight Euclidean 2-designs.

Contribution

It demonstrates the strong non-rigidity of specific tight Euclidean t-designs and offers a complete classification of tight Euclidean 2-designs.

Findings

Certain tight Euclidean t-designs are strongly non-rigid.

There are many non-isomorphic tight Euclidean t-designs for some parameters.

Complete classification of tight Euclidean 2-designs is provided.

Abstract

We study the non-rigidity of Euclidean $t$-designs, namely we study when Euclidean designs (in particular certain tight Euclidean designs) can be deformed keeping the property of being Euclidean $t$-designs. We show that certain tight Euclidean $t$-designs are non-rigid, and in fact satisfy a stronger form of non-rigidity which we call strong non-rigidity. This shows that there are plenty of non-isomorphic tight Euclidean $t$-designs for certain parameters, which seems to have been unnoticed before. We also include the complete classification of tight Euclidean $2$-designs.