On spherical 4-distance 7-designs

TL;DR

This paper studies spherical 4-distance 7-designs, deriving divisibility conditions and providing computational evidence supporting the conjecture that only tight designs exist, culminating in a computer-assisted proof for dimensions up to 1000.

Contribution

It establishes new divisibility conditions for spherical 4-distance 7-designs and proves the conjecture for all dimensions up to 1000 using computer assistance.

Findings

Derived divisibility conditions relating dimension and design size.

Supported the conjecture that only tight designs are possible.

Provided a computer-assisted proof for dimensions up to 1000.

Abstract

We investigate spherical 4-distance 7-designs by studying their distance distributions. We compute these distance distributions and use their product (an integer) to derive certain divisibility conditions relating the dimension $n$ and the cardinality $M$ of our designs. It follows that $n$ divides $12M$ and $n+1$ divides $4M^2$. This result provides a good base for computer experiments to support the folklore conjecture that the only spherical 4-distance 7-designs are the tight spherical 7-designs. We then proceed with a computer assisted proof of this conjecture in all dimensions $n \leq 1000$.