New Spherical 4-Designs(*)

R. H. Hardin and N. J. A. Sloane(**)

Mathematical Sciences Research Center 
AT&T Bell Laboratories,
Murray Hill, New Jersey 07974 USA 


(*) A slightly different version of this paper appeared in
Discrete Math., 106/107 (1992), 255-264.

(**) Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com


Dedicated to J. H. van Lint on the occasion of his 60th birthday


ABSTRACT

This paper gives a number of new spherical 4-designs, and presents
numerical evidence that spherical 4-designs containing n points
in k-dimensional space with k <= 8 exist
precisely for the following values of n and k:
n even and >= 2 for k=1;
n >= 5 for k=2;
n=12, 14, >= 16 for k=3;
n >= 20 for k=4;
n >= 29 for k=5;
n=27, 36, >= 39 for k=6;
n >= 53 for k=7;
and n >= 69 for k=8.


For the full version see
http://NeilSloane.com/doc/spher.pdf (pdf) or
http://NeilSloane.com/doc/spher.ps (ps)