# Database of difference sets with order (n=k-lambda) less than or equal to 15,
# created so as to be convenient for GAP. (Download this as a text file.)

# Omar AbuGhneim and Ken Smith, 
# Last modified April 22, 2006.

# ALL difference sets of order less than 16 occur here.
# There are 95 such difference sets.

# Each line has 3 entries:  the parameters [v,k,lambda],
# the GAP catalogue number [v,cn], and then the index set I
# for which the difference set is 
# { Elements( SmallGroup( v, cn ) )[i]  : i in I }.

# Last modified, May 1, 2006.

#     To run this program, save it as "DifSetDatabase_order_le_15.txt"
# in the same folder as Gap.  Then insert the following 
# Read statement at a GAP command line (after removing the "#gap>"):

# gap>  Read("DifSetDatabase_order_le_15.txt");

# Then, for a given member of the database, say
# gap> DS := DifSetDatabase[ <number> ];
# define the difference set by:
# gap> e := Elements( SmallGroup( DS[2][1], DS[2][2] ) );
# gap> difset := DS[3];

# For example, to look at the first (16,6,2) difference set
# list in the 'modular group' M_{16} = [16,6], below, one 
# might, after reading in this file, run the following 
# GAP commands:
# gap> DS := DifSetDatabase[18];
# gap> e := Elements( SmallGroup( DS[2][1], DS[2][2] ) );
# gap> difset := DS[3];

# Each difference set listed is the "lexicographically least" 
# of all equivalent difference sets.
# (In this ordering, [1,2,4] < [1,2,5] < [1,3,4] < [2,3,4].)

DifSetDatabase := [ 

# Order 2
 
[ [ 7, 3, 1 ], [ 7, 1 ], [ 1, 2, 4 ] ], 


# Order 3
 
[ [ 11, 5, 2 ], [ 11, 1 ], [ 1, 2, 3, 5, 8 ] ], 

[ [ 13, 4, 1 ], [ 13, 1 ], [ 1, 2, 4, 10 ] ], 


# Order 4
 
[ [ 21, 5, 1 ], [ 21, 1 ], [ 1, 2, 3, 9, 10 ]], 
[ [ 21, 5, 1 ], [ 21, 2 ], [ 1, 2, 7, 10, 16 ] ],
 

[ [ 16, 6, 2 ], [ 16, 2 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 2 ], [ 1, 2, 3, 4, 10, 14 ] ],
[ [ 16, 6, 2 ], [ 16, 2 ], [ 1, 2, 3, 8, 9, 11 ] ], 
[ [ 16, 6, 2 ], [ 16, 3 ], [ 1, 2, 3, 4, 10, 14 ] ],
[ [ 16, 6, 2 ], [ 16, 3 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 3 ], [ 1, 2, 3, 7, 10, 11 ] ],
[ [ 16, 6, 2 ], [ 16, 3 ], [ 1, 2, 3, 5, 7, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 4 ], [ 1, 2, 3, 4, 10, 14 ] ],
[ [ 16, 6, 2 ], [ 16, 4 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 4 ], [ 1, 2, 3, 5, 7, 15 ] ],
[ [ 16, 6, 2 ], [ 16, 5 ], [ 1, 2, 3, 4, 11, 13 ] ], 
[ [ 16, 6, 2 ], [ 16, 5 ], [ 1, 2, 3, 4, 8, 15 ] ],
[ [ 16, 6, 2 ], [ 16, 6 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 6 ], [ 1, 2, 3, 8, 9, 11 ] ],
[ [ 16, 6, 2 ], [ 16, 8 ], [ 1, 2, 3, 4, 7, 10 ] ], 
[ [ 16, 6, 2 ], [ 16, 8 ], [ 1, 2, 3, 4, 10, 14 ] ],
[ [ 16, 6, 2 ], [ 16, 9 ], [ 1, 2, 3, 4, 8, 9 ] ], 
[ [ 16, 6, 2 ], [ 16, 9 ], [ 1, 2, 3, 4, 7, 10 ] ],
[ [ 16, 6, 2 ], [ 16, 10 ], [ 1, 2, 3, 4, 5, 16 ] ], 
[ [ 16, 6, 2 ], [ 16, 10 ], [ 1, 2, 3, 4, 8, 15 ] ],
[ [ 16, 6, 2 ], [ 16, 11 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 11 ], [ 1, 2, 3, 5, 7, 15 ] ],
[ [ 16, 6, 2 ], [ 16, 12 ], [ 1, 2, 3, 4, 5, 12 ] ], 
[ [ 16, 6, 2 ], [ 16, 12 ], [ 1, 2, 3, 4, 8, 15 ] ],
[ [ 16, 6, 2 ], [ 16, 13 ], [ 1, 2, 3, 4, 8, 15 ] ], 
[ [ 16, 6, 2 ], [ 16, 13 ], [ 1, 2, 3, 4, 5, 12 ] ],
[ [ 16, 6, 2 ], [ 16, 14 ], [ 1, 2, 3, 4, 5, 16 ] ],
 
[ [ 15, 7, 3 ], [ 15, 1 ], [ 1, 2, 3, 4, 8, 11, 12 ] ],

[ [ 31, 6, 1 ], [ 31, 1 ], [ 1, 2, 4, 9, 13, 19 ] ],
 
# Order 5
 
[ [ 19, 9, 4 ], [ 19, 1 ], [ 1, 2, 3, 4, 6, 8, 13, 14, 17 ] ], 

# Order 6
 
[ [ 23, 11, 5 ], [ 23, 1 ], [ 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18 ] ], 

# Order 7
 
[ [ 57, 8, 1 ], [ 57, 1 ], [ 1, 2, 3, 8, 16, 17, 24, 47 ] ], 
[ [ 57, 8, 1 ], [ 57, 1 ], [ 1, 2, 3, 8, 13, 27, 39, 50 ] ], 
[ [ 57, 8, 1 ], [ 57, 2 ], [ 1, 2, 3, 16, 21, 22, 31, 33 ] ],

[ [ 37, 9, 2 ], [ 37, 1 ], [ 1, 2, 4, 8, 18, 25, 26, 30, 36 ] ],
 

 [ [ 27, 13, 6 ], [ 27, 4 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 16, 19, 20, 23, 26 ] ], 
 [ [ 27, 13, 6 ], [ 27, 4 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 13, 15, 18, 19, 23 ] ], 
 [ [ 27, 13, 6 ], [ 27, 5 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 23, 25, 27 ] ],
 
# Order 8
 
[ [ 73, 9, 1 ], [ 73, 1 ], [ 1, 2, 4, 8, 16, 32, 37, 55, 64 ] ], 
 
[ [ 31, 15, 7 ], [ 31, 1 ], [ 1, 2, 3, 4, 6, 7, 9, 12, 13, 19, 20, 21, 24, 28, 30 ] ], 
[ [ 31, 15, 7 ], [ 31, 1 ], [ 1, 2, 3, 4, 6, 7, 9, 10, 14, 17, 22, 23, 24, 26, 28 ] ],
 
# Order 9
 
[ [ 91, 10, 1 ], [ 91, 1 ], [ 1, 2, 8, 29, 32, 44, 52, 66, 84, 88 ] ], 

[ [ 45, 12, 3 ], [ 45, 2 ], [ 1, 2, 4, 5, 9, 10, 16, 23, 24, 26, 36, 42 ] ], 
[ [ 45, 12, 3 ], [ 45, 2 ], [ 1, 2, 4, 5, 9, 10, 16, 18, 23, 36, 37, 39 ] ], 

[ [ 40, 13, 4 ], [ 40, 1 ], [ 1, 2, 3, 4, 5, 8, 9, 18, 23, 26, 28, 30, 38 ] ], 
[ [ 40, 13, 4 ], [ 40, 2 ], [ 1, 2, 3, 4, 5, 12, 22, 24, 25, 28, 30, 35, 40 ] ], 

[ [ 36, 15, 6 ], [ 36, 8 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 21, 22, 33, 34, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 8 ], [ 1, 2, 3, 4, 5, 6, 10, 12, 14, 16, 18, 21, 22, 35, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 8 ], [ 1, 2, 3, 4, 5, 6, 10, 12, 14, 18, 21, 25, 30, 31, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 8 ], [ 1, 2, 3, 4, 6, 10, 12, 13, 14, 20, 21, 25, 27, 28, 31 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 7, 9, 11, 12, 16, 20, 23, 32, 33, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 9, 11, 12, 15, 16, 17, 20, 23, 33, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 7, 10, 11, 13, 16, 19, 22, 32, 33, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 10, 11, 13, 15, 16, 17, 19, 22, 33, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 6, 7, 14, 17, 20, 21, 22, 28, 30, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 6 ], [ 1, 2, 3, 4, 5, 6, 14, 15, 20, 21, 22, 26, 30, 34, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 7 ], [ 1, 2, 3, 4, 5, 6, 7, 10, 16, 21, 22, 23, 25, 31, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 9 ], [ 1, 2, 3, 4, 5, 7, 12, 13, 16, 20, 23, 24, 25, 26, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 9 ], [ 1, 2, 3, 4, 5, 6, 7, 12, 13, 16, 20, 23, 32, 33, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 9 ], [ 1, 2, 3, 4, 5, 7, 12, 13, 14, 15, 16, 20, 23, 35, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 9 ], [ 1, 2, 3, 4, 11, 14, 17, 24, 25, 29, 30, 32, 33, 34, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 9 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 18, 22, 25, 29, 30, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 14 ], [ 1, 2, 3, 4, 8, 11, 14, 15, 18, 20, 24, 25, 26, 35, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 14 ], [ 1, 2, 3, 4, 5, 7, 10, 14, 16, 21, 22, 23, 26, 31, 32 ] ], 
[ [ 36, 15, 6 ], [ 36, 14 ], [ 1, 2, 3, 4, 5, 6, 7, 10, 16, 21, 22, 23, 25, 31, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 13 ], [ 1, 2, 3, 4, 5, 6, 7, 10, 16, 21, 22, 23, 25, 31, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 17, 19, 23, 32, 33, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 9, 12, 13, 14, 15, 17, 19, 23, 34, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 12, 17, 21, 22, 32, 33, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 10, 11, 12, 14, 15, 17, 21, 22, 34, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 6, 7, 15, 16, 20, 22, 23, 26, 31, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 12 ], [ 1, 2, 3, 4, 5, 7, 14, 16, 20, 22, 23, 25, 31, 33, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 5, 6, 11, 12, 15, 17, 22, 26, 29, 30, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 5, 11, 12, 14, 17, 22, 25, 29, 30, 33, 34 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 18, 19, 22, 23, 25, 31, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 9, 10, 11, 17, 24, 29, 30, 32, 34, 35, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 5, 6, 7, 10, 16, 21, 22, 23, 25, 31, 36 ] ], 
[ [ 36, 15, 6 ], [ 36, 10 ], [ 1, 2, 3, 4, 5, 7, 10, 14, 16, 21, 22, 23, 26, 31, 32 ] ], 
[ [ 36, 15, 6 ], [ 36, 11 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 19, 26, 28, 32, 35 ] ], 
[ [ 36, 15, 6 ], [ 36, 11 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 18, 19, 26, 29, 30, 31, 34 ]], # New!
[ [ 36, 15, 6 ], [ 36, 11 ], [ 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 26, 31, 32, 33, 34 ] ], 

[ [ 35, 17, 8 ], [ 35, 1 ], [ 1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 18, 19, 26, 28, 32, 33, 35 ] ], 

# Order 11
 
[ [ 133, 12, 1 ], [ 133, 1 ], [ 1, 2, 3, 26, 28, 38, 43, 54, 57, 75, 95, 113 ] ], 

[ [ 43, 21, 10 ], [ 43, 1 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 12, 14, 17, 20, 21, 23, 25, 26, 30, 36, 37, 38, 40 ] ], 
[ [ 43, 21, 10 ], [ 43, 1 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 21, 26, 27, 28, 32, 33, 34, 36, 37, 41 ] ],


# Order 12
 
[ [ 47, 23, 11 ], [ 47, 1 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42 ] ], 


# Order 13

[ [ 183, 14, 1 ], [ 183, 1 ], [ 1, 2, 3, 8, 12, 22, 30, 54, 69, 70, 92, 113, 130, 150 ] ], 
[ [ 183, 14, 1 ], [ 183, 1 ], [ 1, 2, 3, 8, 13, 15, 22, 42, 79, 107, 111, 135, 160, 165 ] ], 
[ [ 183, 14, 1 ], [ 183, 2 ], [ 1, 2, 3, 18, 26, 32, 39, 51, 85, 114, 131, 141, 142, 151 ] ], 
 

# Order 15

[ [ 59, 29, 14 ], [ 59, 1 ], [ 1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 18, 21, 23, 25, 26, 27, 29, 32, 36, 38, 40, 41, 43, 44, 48, 53, 54 ] ]

];