24 points in 10-D with secant 10.  Found by wds and njas circa April 1990.

wds, rhh, Here is a beautiful code equivalent to the one warren found
with 24 vectors in 10-D and secant 10. There are two versions here.

Version 1: each vector is described by 14 real numbers,
its coords in 14-space, one number per line. Thus there are 336
lines
To verify that this has the correct min angle , feed it into 
secsum AS IF IT WERE A 14-D CODE.
(Give it a file name temp.14.24 in other words)

To verify that it really is a 10-D code, note that the first
12 coords come in 4 sets of 3, and each set of 3 sums to zero.
So the first 12 coords are really only in an 8-D space.
This can be seen more easily in the second version.

Of course if you really want 10 coords, just do this to each triple:
x -x 0  -> (x, 0 )*sqrt(2)
0 x -x  -> (-x/2, +x*sqrt(3)/2 )*sqrt(2)
-x 0 x  -> (-x/2, -x*sqrt(3)/2 )*sqrt(2)

Version 2 is a symbolic description of the EXACTLY the same vectors
This is at the end of the file.  This shows how pretty
the code is and explains its construction.

Cheers!  Neil



Version 1
2.121320343560
-2.121320343560
0
0
0
0
0
0
0
0
0
0
1
0
0
2.121320343560
-2.121320343560
0
0
0
0
0
0
0
0
0
1
0
-2.121320343560
0
2.121320343560
0
0
0
0
0
0
0
0
0
1
0
0
0
0
2.121320343560
-2.121320343560
0
0
0
0
0
0
0
1
0
0
0
0
0
2.121320343560
-2.121320343560
0
0
0
0
0
0
1
0
0
0
0
-2.121320343560
0
2.121320343560
0
0
0
0
0
0
1
0
0
0
0
0
0
0
2.121320343560
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0
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1
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0
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0
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2.121320343560
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0
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0
-2.121320343560
0
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0
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0
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0
1.732050807568877
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0
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0
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-1.25
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0
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-.86602540378443864676
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Version 2:
Symbolic version:
10 24
a -a 0 0 0 0 0 0 0 0 0 0 t 0
0 a -a 0 0 0 0 0 0 0 0 0 t 0
-a 0 a 0 0 0 0 0 0 0 0 0 t 0
0 0 0 a -a 0 0 0 0 0 0 0 t 0
0 0 0 0 a -a 0 0 0 0 0 0 t 0
0 0 0 -a 0 a 0 0 0 0 0 0 t 0
0 0 0 0 0 0 a -a 0 0 0 0 t 0
0 0 0 0 0 0 0 a -a 0 0 0 t 0
0 0 0 0 0 0 -a 0 a 0 0 0 t 0
0 0 0 0 0 0 0 0 0 u -u 0 t +u
0 0 0 0 0 0 0 0 0 0 u -u t +u
0 0 0 0 0 0 0 0 0 -u 0 u t +u
0 0 0 0 0 0 0 0 0 -u u 0 t -u
0 0 0 0 0 0 0 0 0 0 -u u t -u
0 0 0 0 0 0 0 0 0 u 0 -u t -u
 0 -c c  0 -c c  0 -c c  0 -d d  e f
 0 -c c  c 0 -c  c 0 -c  d 0 -d  e f
 0 -c c  -c c 0  -c c 0  -d d 0  e f
 c 0 -c  c 0 -c  -c c 0  0 -d d  e f
 -c c 0  -c c 0  c 0 -c  0 -d d  e f
 c 0 -c  -c c 0  0 -c c  d 0 -d  e f
 -c c 0  c 0 -c  0 -c c  -d d 0  e f
 c 0 -c  0 -c c  c 0 -c  -d d 0  e f
 -c c 0  0 -c c  -c c 0  d 0 -d  e f
 
where a=3/sqrt(2), t=1, u=sqrt(3), c=3/sqrt(8), d=sqrt(3)/2,
e=-5/4, f=sqrt(3)/4.  The last 9 rows are the tetracode
which is the 9-word linear code over GF(3) generated by
0111 and 1120,  with 0 replaced by 0 c -c (or 0 d -d),
1 replaced by -c 0 c, and 2 by c -c 0.

Description:
The first 9 points are the join of 3 equilateral triangles
in 3 perpendicular 2-spaces.

The next 6 points are a triangle and its negative in another
perpendicular 2-space.

The last 9 points are equilateral triangles at a different scale
in these 4 2-spaces, the particular triangles used being
determined by the tetracode.

The last 2 coords in each vector specify 2 stacking coords
to keep the layers apart.


The group has order 108.

  min norm =   1.80000000  between  1 16
  min angle =   8.426082952e+01  between  1 16
  secant =  10.00000000