An integer function of finite sets of points on spheres

[Paul C. Leopardi]

Hi all,
Here is an integer function of a set of points on a sphere, which can be used 
to generate integer sequences from sequences of point sets on spheres.

In brief, in Matlab:
function R = rsg(x)
%rsg The number of distinct sets of inner products of x
R = rank(sort(x'*x));

In detail:
Take a set of N points on the sphere S^d in R^(d+1), expressed in Cartesian 
coordinates as the (d+1) by N matrix x.
Now, using (eg.) Matlab, calculate

R = rank(sort(x'*x));
ie.,
M = x'*x;
Y = sort(M);
R = rank(Y);

The expression M = x'*x gives the usual Gram matrix of inner products of the 
points of x. This is symmetric by construction.

The sort function Y = sort(M) (at least in Matlab) sorts the columns of the 
matrix M, so that each column Y(:,k) of Y contains the inner products of x 
with respect to x(k) sorted in increasing order.

The rank function R = rank(Y) then determines the number of different columns 
of Y, which is the number of different sets of distinct inner products of x.
Since R is a rank of a non-zero N*N matrix, it must be an integer between 1 
and N inclusive. The ratio R/N could be used to measure a kind of geometric 
irregularity. If R == 1, ie. R/N == 1/N, then the set of points would be very 
regular in this sense and if R == N, ie. R/N == 1, the set of points would be 
very irregular.

Example:
When used on Hardin and Sloane's Library of 3-d designs
http://www.research.att.com/~njas/sphdesigns/dim3
for spherical designs with N=(n+1)^2 points, the function rsg yields the 
following sequences N, R:
n: 1, 2,  3,  4,  5,  6,  7, ...
N: 4, 9, 16, 25, 36, 49, 64, ...
R: 1, 2,  2,  3,  3, 48, 13, ...

Notice the large contrast between R(5)/N(5) == 1/12 and R(6)/N(6) == 48/49.

I have not yet calculated the rsg of all of the point sets in the library of 
3-d designs, because it takes too long to download each file individually. Is 
there somewhere I can download a compressed (eg. Gzipped) file containing all 
of these point sets?
Best regards

References:
"Uniqueness of the 120-point spherical 11-design in four dimensions", Archiv 
der Mathematik, P. Boyvalenkov and D. Danev, Volume 77, Number 4,
October 2001, 360 - 368. 
http://www.springerlink.com/link.asp?id=vkuwt9wda3wj0jnv

"McLaren's Improved Snub Cube and Other New Spherical Designs in Three 
Dimensions", R. H. Hardin and N. J. A. Sloane,  Discrete and Computational 
Geometry, 15 (1996), pp. 429-441. 
http://www.research.att.com/~njas/sphdesigns/

Matlab documentation on x'*x, sort and rank
http://www.mathworks.com/access/helpdesk/help/techdoc/ref/arithmeticoperators.html
http://www.mathworks.com/access/helpdesk/help/techdoc/ref/sort.html
http://www.mathworks.com/access/helpdesk/help/techdoc/ref/rank.html