Parallel Combinatorics

A) Perpendicular Arrays

B) Steiner Triple Systems

A)  Entries so that no pairs of columns contain the same two entries.

PA 5 3
 1  2  3  
 1  3  4  
 2  3  5  
 2  4  1  
 3  4  2  
 3  5  4  
 4  1  5  
 4  5  2  
 5  1  3  
 5  2  1  


PA 7 5
 1   2   3   4   5 
 1   3   4   2   6 
 1   4   2   3   7 
 2   3   5   7   4 
 2   5   6   3   1 
 2   7   3   5   6 
 3   4   6   5   2 
 3   5   4   6   7 
 3   6   7   4   1 
 4   2   7   6   3 
 4   6   2   7   5 
 4   7   5   2   1 
 5   1   3   7   2 
 5   4   7   1   6 
 5   7   1   4   3 
 6   1   4   5   3 
 6   2   5   1   7 
 6   5   1   2   4 
 7   1   2   6   4 
 7   3   6   1   5 
 7   6   1   3   2

B) A sample Steiner solution
S(2,3,13) system 

the final solution is:  
  1  2  3
  1  4  5
  1  6  7
  1  8  9
  1 10 11
  1 12 13
  2  4  6
  2  5  7
  2  8 10
  2  9 12
  2 11 13
  3  4  8
  3  5  9
  3  6 10
  3  7 13
  3 11 12
  4  7 11
  4  9 13
  4 10 12
  5  6 12
  5  8 11
  5 10 13
  6  8 13
  6  9 11
  7  8 12
  7  9 10