### This work done by Omar A. AbuGhneim from Jordan University (Email: o_abughneim@yahoo.com). # There are 267 groups of order 64. Among them, there are 259 groups that # admit (64,28,12) difference sets. The other 8 groups do not have any # (64,28,12) difference sets. We wrote programs using the software GAP # to construct all (64,28,12) difference sets in the 259 groups of order # 64 that admit (64,28,12) difference sets. # We refer to the groups as they appear in the SmallGroups library of # the software package GAP. For instance, when we work in groups of # order 64 and write [64,14] we mean group number 14 of order 64 in the # GAP library. # The 8 groups that do not admit (64,28,12) difference sets are # [64, 1], [64, 38], [64, 47], [64, 50], [64, 52], [64, 53], [64, 54], # and [64, 186]. Indeed, these groups are the ones that have either # the cyclic group of order 32 as a factor group or have the dihedral # group of order 32 as a factor group. # Also, we find all (64,28,12) difference sets that are reversible # i.e. all (64,28,12) difference sets D with D=D^(-1). There are 184 # groups of order 64 that have reversible (64,28,12) difference sets. # These groups are [64,i] where i in {2, 3, 4, 5, 8, 9, 10, 11, 12, 13, # 15, 16, 18, 20, 22, 23, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 37, 40, # 41, 42, 43, 45, 46, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, # 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, # 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, # 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 126, # 127, 128, 129, 130, 131, 133, 134, 135, 138, 140, 141, 142, 144, 145, 146, # 147, 149, 150, 151, 152, 153, 155, 156, 157, 159, 160, 161, 162, 163, 164, # 166, 167, 169, 170, 172, 173, 174, 177, 178, 179, 182, 184, 187, 190, 192, # 193, 194, 195, 196, 197, 199, 200, 202, 203, 204, 205, 206, 207, 208, 209, # 210, 211, 213, 214, 215, 216, 219, 221, 222, 226, 227, 231, 232, 234, 236, # 240, 241, 242, 245, 247, 249, 250, 251, 254, 260, 261, 263, 267 }. # We used the software GAP to find all (64,28,12) difference sets and # reversible (64,28,12) difference sets. # In this file we list all inequivalent (64,28,12) difference sets # in group [64,189]. Also, we determine the nonisomorphic # symmetric (64,28,12) designs that rise from these DSs. # Note that there is no reversible (64,28,12) difference sets in # group [64,189]. # To determine the nonisomorphic symmetric (64,28,12) designs, we used # the DESIGN package for GAP, Version 1.6, 2011, by L.H. Soicher, # http://designtheory.org/software/gap design/. # Note: The GAP command 'e:=Elements(SmallGroup(64,189));' can be # used to create a list e of the elements of the group [64,189]. # In our list of (64,28,12) DSs in group [64,189], we have [1, # 2, 3, 4, 5, 6, 8, 10, 15, 18, 21, 23, 27, 28, 29, # 31, 32, 33, 35, 39, 48, 50, 53, 54, 55, 57, 60, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[10], e[15], e[18], e[21], e[23], e[27], e[28], e[29], e[31], # e[32], e[33], e[35], e[39], e[48], e[50], e[53], e[54], e[55], e[57], e[60], e[63]] # where e is the list of the elements of the group [64,189] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,189));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 189] ### The number of inequivalent (64,28,12) DSs in group [64,189] is 104 ### The list of all inequivalent (64,28,12) DSs in group [64,189] is DS189:=[ [ 1, 2, 3, 4, 5, 6, 8, 10, 15, 18, 21, 23, 27, 28, 29, 31, 32, 33, 35, 39, 48, 50, 53, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 17, 18, 21, 24, 25, 26, 28, 30, 35, 40, 43, 48, 50, 51, 55, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 20, 21, 23, 26, 28, 32, 35, 37, 38, 41, 42, 43, 45, 47, 53, 55, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 17, 18, 20, 26, 27, 29, 33, 34, 35, 37, 38, 39, 44, 49, 50, 53, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18, 21, 23, 27, 29, 31, 33, 35, 39, 48, 49, 52, 53, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18, 20, 21, 23, 24, 27, 29, 31, 35, 42, 47, 48, 49, 52, 55, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 17, 18, 25, 26, 29, 33, 35, 36, 37, 42, 43, 46, 48, 55, 57, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 17, 18, 24, 26, 29, 33, 35, 36, 37, 42, 44, 46, 47, 55, 57, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 17, 18, 24, 25, 26, 29, 33, 35, 36, 37, 42, 47, 48, 55, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 17, 18, 20, 26, 27, 29, 33, 35, 37, 39, 44, 49, 50, 53, 55, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 18, 20, 21, 23, 25, 27, 29, 31, 34, 35, 37, 38, 42, 44, 49, 52, 53, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 18, 21, 23, 27, 29, 31, 35, 36, 37, 39, 48, 49, 52, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 18, 20, 21, 23, 27, 29, 31, 35, 37, 42, 48, 49, 52, 53, 55, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 26, 29, 35, 36, 42, 43, 44, 54, 55, 57, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 25, 26, 29, 35, 36, 42, 43, 46, 48, 54, 55, 57, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 24, 26, 29, 35, 36, 42, 44, 46, 47, 54, 55, 57, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 24, 25, 26, 29, 35, 36, 42, 47, 48, 54, 55, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 20, 26, 29, 35, 39, 43, 44, 53, 54, 55, 56, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 20, 26, 27, 29, 35, 39, 44, 49, 50, 53, 54, 55, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 20, 24, 26, 29, 35, 39, 44, 46, 47, 53, 54, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 18, 20, 24, 25, 26, 29, 35, 39, 47, 48, 53, 54, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 20, 23, 26, 27, 31, 33, 34, 39, 40, 41, 45, 51, 52, 53, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 20, 23, 26, 27, 31, 34, 36, 37, 39, 40, 41, 45, 51, 52, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16, 20, 23, 26, 27, 31, 37, 38, 39, 40, 41, 45, 51, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 20, 21, 26, 27, 28, 31, 32, 35, 38, 41, 42, 43, 44, 47, 50, 55, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 20, 21, 23, 26, 28, 31, 32, 35, 38, 41, 42, 43, 45, 47, 49, 55, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 20, 21, 25, 27, 28, 31, 32, 33, 34, 36, 37, 41, 42, 44, 45, 50, 53, 55 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 20, 21, 23, 27, 28, 31, 32, 34, 37, 41, 42, 44, 48, 50, 53, 54, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 17, 25, 26, 28, 32, 34, 39, 41, 42, 46, 48, 50, 52, 55, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 17, 24, 25, 26, 28, 32, 34, 39, 41, 42, 43, 47, 48, 50, 52, 55, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 17, 23, 26, 28, 32, 33, 34, 39, 41, 42, 45, 50, 52, 53, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 17, 23, 24, 26, 28, 32, 34, 39, 41, 42, 45, 46, 47, 50, 52, 55, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 17, 23, 26, 28, 32, 33, 36, 38, 39, 41, 42, 45, 50, 52, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 16, 17, 23, 26, 28, 32, 34, 36, 37, 39, 41, 42, 45, 50, 52, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 16, 17, 23, 26, 28, 32, 37, 38, 39, 41, 42, 45, 50, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 17, 25, 26, 28, 32, 35, 39, 41, 42, 43, 46, 48, 50, 52, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 17, 24, 26, 28, 32, 35, 39, 41, 42, 44, 46, 47, 50, 52, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 17, 24, 25, 26, 28, 32, 35, 39, 41, 42, 47, 48, 50, 52, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 20, 21, 23, 26, 28, 31, 32, 41, 42, 45, 46, 47, 49, 55, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 15, 17, 21, 25, 26, 28, 31, 32, 40, 41, 46, 47, 48, 49, 55, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 20, 25, 26, 29, 31, 35, 36, 37, 38, 39, 40, 41, 48, 49, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 18, 20, 21, 26, 29, 30, 34, 35, 38, 42, 44, 47, 50, 59, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 17, 18, 20, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 44, 52, 53, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 17, 18, 20, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 48, 49, 53, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 17, 18, 20, 25, 26, 29, 31, 33, 34, 35, 36, 37, 38, 39, 48, 49, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 20, 21, 25, 29, 34, 36, 37, 41, 42, 44, 45, 50, 52, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 20, 21, 25, 27, 29, 30, 33, 34, 37, 41, 42, 44, 45, 55, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 20, 21, 23, 27, 29, 30, 34, 36, 37, 41, 42, 44, 48, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 21, 23, 29, 30, 33, 34, 36, 37, 42, 48, 50, 55, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 21, 23, 25, 29, 33, 34, 36, 37, 42, 44, 49, 50, 52, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 21, 23, 25, 29, 30, 33, 34, 37, 42, 44, 50, 53, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, 21, 24, 26, 29, 34, 35, 41, 42, 44, 46, 50, 52, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 24, 25, 26, 29, 30, 35, 39, 43, 48, 50, 55, 57, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 23, 25, 29, 33, 35, 39, 44, 49, 50, 52, 53, 54, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 23, 24, 25, 29, 35, 39, 44, 46, 47, 49, 50, 52, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 24, 26, 29, 35, 42, 43, 44, 46, 49, 50, 52, 55, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 24, 26, 29, 30, 35, 42, 43, 44, 50, 55, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 23, 25, 29, 35, 42, 43, 44, 46, 49, 50, 52, 55, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 23, 25, 29, 33, 35, 36, 42, 44, 49, 50, 52, 54, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 23, 24, 25, 29, 35, 42, 44, 47, 49, 50, 52, 55, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 18, 26, 29, 30, 31, 33, 35, 36, 37, 42, 44, 52, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 18, 25, 26, 29, 31, 33, 35, 36, 37, 42, 48, 49, 55, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 18, 20, 26, 29, 30, 31, 33, 35, 37, 39, 44, 52, 53, 55, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 18, 20, 25, 26, 29, 31, 33, 35, 37, 39, 48, 49, 53, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 21, 23, 27, 29, 30, 36, 37, 38, 39, 41, 44, 48, 54, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 21, 23, 25, 29, 30, 33, 37, 38, 39, 44, 50, 53, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 26, 29, 30, 31, 35, 36, 38, 39, 41, 42, 44, 52, 53, 54, 55, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 25, 26, 29, 31, 35, 36, 38, 39, 41, 42, 48, 49, 53, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 26, 29, 30, 31, 34, 35, 38, 42, 44, 52, 53, 54, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 26, 29, 30, 31, 34, 35, 36, 38, 42, 44, 52, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 25, 26, 29, 31, 34, 35, 38, 42, 48, 49, 53, 54, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 17, 18, 25, 26, 29, 31, 34, 35, 36, 38, 42, 48, 49, 54, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 20, 21, 23, 29, 34, 36, 41, 42, 44, 48, 50, 52, 53, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 23, 29, 34, 36, 42, 48, 49, 50, 52, 54, 55, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 20, 26, 29, 30, 31, 34, 35, 39, 40, 41, 44, 52, 54, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 20, 25, 26, 29, 31, 34, 35, 39, 40, 41, 48, 49, 54, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 21, 23, 25, 29, 35, 36, 37, 39, 44, 49, 50, 52, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 23, 25, 29, 35, 37, 42, 44, 49, 50, 52, 53, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 26, 29, 30, 31, 33, 34, 35, 36, 39, 41, 42, 44, 52, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 25, 26, 29, 31, 33, 34, 35, 36, 39, 41, 42, 48, 49, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 18, 26, 29, 30, 31, 35, 36, 42, 44, 52, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 18, 25, 26, 29, 31, 35, 36, 42, 48, 49, 54, 55, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 18, 20, 26, 29, 30, 31, 35, 39, 44, 52, 53, 54, 55, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 18, 20, 25, 26, 29, 31, 35, 39, 48, 49, 53, 54, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 21, 23, 29, 36, 38, 39, 41, 44, 48, 50, 52, 53, 54, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 23, 29, 36, 38, 39, 48, 49, 50, 52, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 21, 25, 26, 27, 29, 30, 34, 38, 39, 41, 43, 47, 48, 57, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 20, 26, 29, 30, 31, 34, 37, 38, 39, 40, 41, 44, 52, 53, 54, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 20, 25, 26, 29, 31, 34, 37, 38, 39, 40, 41, 48, 49, 53, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 20, 21, 26, 27, 29, 30, 34, 38, 41, 42, 43, 44, 47, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 21, 25, 27, 29, 39, 41, 43, 44, 45, 49, 52, 55, 57, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 21, 25, 27, 29, 33, 39, 41, 44, 45, 49, 52, 53, 54, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 21, 24, 25, 27, 29, 39, 41, 44, 45, 46, 47, 49, 52, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 52, 55, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 25, 26, 29, 31, 33, 37, 39, 40, 41, 48, 49, 55, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 25, 27, 29, 41, 42, 43, 44, 45, 46, 49, 52, 55, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 25, 27, 29, 33, 36, 41, 42, 44, 45, 49, 52, 54, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 24, 25, 27, 29, 41, 42, 44, 45, 47, 49, 52, 55, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 17, 26, 29, 30, 31, 36, 37, 39, 41, 42, 44, 52, 54, 55, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 17, 25, 26, 29, 31, 36, 37, 39, 41, 42, 48, 49, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 17, 26, 29, 30, 31, 34, 38, 39, 41, 42, 44, 52, 54, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 17, 25, 26, 29, 31, 34, 38, 39, 41, 42, 48, 49, 54, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 21, 25, 27, 29, 36, 37, 39, 41, 44, 45, 49, 52, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 20, 21, 25, 27, 29, 37, 41, 42, 44, 45, 49, 52, 53, 55, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 70 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS189[1], DS189[2], DS189[3], DS189[5], DS189[6], DS189[11], DS189[12], DS189[13], DS189[14], DS189[16], DS189[17], DS189[21], DS189[23], DS189[25], DS189[26], DS189[27], DS189[28], DS189[31], DS189[32], DS189[35], DS189[37], DS189[39], DS189[40], DS189[42], DS189[44], DS189[46], DS189[47], DS189[48], DS189[49], DS189[50], DS189[51], DS189[52], DS189[53], DS189[54], DS189[55], DS189[56], DS189[57], DS189[58], DS189[59], DS189[60], DS189[61], DS189[62], DS189[63], DS189[64], DS189[65], DS189[66], DS189[67], DS189[69], DS189[73], DS189[74], DS189[77], DS189[78], DS189[79], DS189[81], DS189[82], DS189[85], DS189[86], DS189[87], DS189[90], DS189[91], DS189[92], DS189[93], DS189[95], DS189[96], DS189[97], DS189[98], DS189[100], DS189[102], DS189[103], DS189[104]. # Note that there is no reversible (64,28,12) difference sets in # group [64,189]. ########################### ########################### ###########################