This table is a companion to the paper
'Paving Small matrices and the Kadison-Singer Extension Problem II - Computational Results'
by Dieter Schmidt, Gary Weiss, and Vrej Zarikian, to appear.
The values listed here are updated when better results are found. 
The links given with a paving value leads to the corresponding matrix in MATLAB format, that is to a MATLAB file with the extension .mat.
 In order to view the matrix it is necessary to download the file and then to load it into MATLAB.
More explanations are found at the bottom of the page.
3-Paving Parameters
Size  General Real Circulant Self-Adjoint Real-Symmetric SA Circulant Upper-Triangular Non-Negative Toeplitz Size 
4 .6180# .6180# .6000# .5774# .4472# .4142# .5412# .5550# .6180# 4
5 .6180# .6180# .6120? ≤ .6180 .5774# .4472# .4472# .5609? ≤ .5774 .5550# .6124? ≤ .6180 5
6 .7071# .7071# .5726? ≤ .6325 .5774# .4851# .4069? ≤ .4495 .5725? ≤ .5774 .5550? ≤ .5574* .6258? ≤ .7071 6
7 .8239? < 1 .8029? < 1 .8239? < 1 .6872? < .7559 .6667? < .7559 .6544? < .7559 .6503? ≤ .9258 " ≤ .6667 .8239? 7
8 .8239? " .7651? ≤ .9258 " ≤ .8819 " ≤ .8819 .5797? ≤ .7559 .6599? " ≤ .6667 .7651? 8
9 .8238? " .7177? ≤ .8165 " ≤ .8889 " ≤ .8889 .5539? ≤ .6667 .6628? " ≤ .6667 .7177? 9
10 .8540? .8387? .8540? .7536? .7454? .6686? .6703? " ≤ .6667 .8540? 10
11 " " .8089? " " .6230? ≤ .9045 " " ≤ .6667 .8089? 11
12 " " .7639? " " .6574? " " ≤ .6667 .7691? 12
13 .8615? " .8615? " " .6983? .6852? " ≤ .6667 .8615? 13
14 " " .8239? kron(A7,I2) " " .6726? " " ≤ .6667 .8239? kron(A7,I2) 14
15 " " .7903? " " .6693? " " ≤ .6667 .7903? 15
16 " " .8523? .7574? " .7019? .6940? " ≤ .6667 .8523? 16
17 " " .8125? " " ? " " ≤ .6667 .8125? 17
18 " " .7617? " " .5539? kron(A9,I2) " " ≤ .6667 .7617? 18
19 " " .8424? " " .6767? " " ≤ .6667 .8424? 19
20 " " .8540? kron(A10,I2) " " .6686? kron(A10,I2) " " ≤ .6667 .8540? kron(A10,I2) 20
21 " " .8239? kron(A7,I3) " " .6544? kron(A7,I3) " " ≤ .6667 .8239? kron(A7,I3) 21
22 " " .8230? " " .6230? kron(A11,I2) " " ≤ .6667 .8230? 22
23 " " ? " " ? " " ≤ .6667 ? 23
24 " " .7691? kron(A12,I2) " " .6574? kron(A12,I2) " " ≤ .6667 .7691? kron(A12,I2) 24
25 " " .8328? " " .4472? kron(A5,I5) " " ≤ .6667 .8328? 25
Explanations of symbols used in table
#  The value has been proven to be the best possible
?   The matrix with this value has been found numerically, but may not be the best possible. If an inequality is given to the right of it in red, then this a proven upper bound. 
" Means that the value from the previous line can be obtained by taking the direct sum of that matrix with a block of 0's.  
The corresponding MATLAB command would be  blkdiag(A,zeros(m))  with m=1,2,3,...  
A related construction is the tensor product of two matrices A and B.  The corresponding MATLAB function is kron(A,B).  
kron(Am,In) in our notation means to take the matrix in the same category of size m and form the tensor product with the identity matrix of size n
* unpublished ≤ 0.5577, .5774 in WZ
2-Paving Parameters
Size General Real Circulant Self-Adjoint Real-Symmetric SA Circulant Upper-Triangular Non-Negative Toeplitz Size
3 1# 1# 1# .5774# .5000# .5774# .6180# 1# 1# 3
4 " " .6000# .5774# .5493? ≤ .5577 .4142? ≤ .5774 .7071# " .6000? 4
5 " " 1# .8944# .8944# .8944# .7715? " 1# 5
6 " " 1# .8944# .8944# .7454? ≤ .8944 .8337? " 1# 6
7 " " 1# .9225? " .9073? .8500? " 1# 7
8 " " .9623? .9225? " .7701? .8866? " .9623? 8
9 " " 1# .9414? " .8920? .8965? " 1# 9
10 " " 1# .9414? " .8944? .9149? " 1# 10
11 " " 1# .9477? " .9315? .9207? " 1# 11
12 " " 1# .9477? " .8638? " " 1# 12
13 " " 1# .9547? " .9049? " " 1# 13
14 " " 1# .9547? " .9381? " " 1# 14
15 " " 1# .9625? " .9260? " " 1# 15
16 " " .9846? .9625? " .8778? " " .9846? 16
17 " " 1# .9692? " .9192? " " 1# 17
18 " " 1# .9692? " .9107? " " 1# 18
19 " " 1# .9742? " .9229? " " 1# 19
20 " " 1# .9742? " .8944? " " 1# 20
21 " " 1# .97618? " .9073? " " 1# 21
22 " " 1# .97620? " .9315? " " 1# 22
23 " " 1# .9780? " .9292? " " 1# 23
24 " " 1# " " .8638? " " 1# 24
25 " " 1# " " .8944? " " 1# 25
References:
1 K. Berman, H. Halpern, V. Kaftal and G. Weiss, Matrix norm inequalities and the relative Dixmier property,  Integral Equations Operator Theory 11 (1988), no. 1, 28-48.  
2 G. Weiss and V. Zarikian, Paving Small Matrices and the Kadison-Singer Extension Problem, Operators and Matrices, Vol. 4, no. 3, (2010), 301-352. 
3 D. Schmidt, G. Weiss and V. Zarikian, Paving Small Matrices and the Kadison-Singer Extension Problem II - Computational Results,  SCIENCE CHINA Mathematics: Kadison Proceedings, to appear. 
4 Cates and V.Zarikian,