### 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 [64,19]. 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 [64,19]. # 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(64,19));' can be # used to create a list e of the elements of the group [64,19]. # In our list of (64,28,12) DSs in group [64,19], we have [1, # 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 24, 25, # 26, 27, 28, 31, 35, 40, 47, 48, 49, 52, 57, 58, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[9], e[11], e[13], e[16], e[18], e[20], e[24], e[25], e[26], # e[27], e[28], e[31], e[35], e[40], e[47], e[48], e[49], e[52], e[57], e[58], e[64]] # where e is the list of the elements of the group [64,19] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,19));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 19] ### The number of inequivalent (64,28,12) DSs in group [64,19] is 176 ### The list of all inequivalent (64,28,12) DSs in group [64,19] is DS19:=[ [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 24, 25, 26, 27, 28, 31, 35, 40, 47, 48, 49, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 24, 25, 26, 27, 28, 30, 32, 35, 39, 40, 42, 47, 48, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 17, 20, 26, 29, 30, 31, 34, 35, 36, 41, 44, 52, 53, 54, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 17, 20, 26, 28, 29, 31, 34, 36, 37, 38, 41, 46, 51, 54, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 17, 20, 26, 29, 30, 31, 33, 34, 38, 41, 44, 52, 53, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 17, 20, 26, 28, 29, 31, 33, 34, 38, 41, 46, 51, 53, 56, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 26, 29, 30, 31, 33, 34, 36, 39, 41, 42, 44, 52, 53, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 26, 28, 29, 31, 34, 39, 41, 42, 46, 51, 53, 54, 56, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 18, 26, 27, 36, 38, 39, 40, 42, 44, 49, 51, 52, 53, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 18, 26, 27, 34, 38, 39, 40, 42, 46, 49, 51, 52, 53, 55, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 14, 15, 16, 18, 26, 27, 34, 37, 38, 39, 40, 42, 43, 49, 51, 52, 55, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 16, 18, 26, 27, 28, 30, 33, 35, 39, 40, 42, 44, 49, 53, 54, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 16, 18, 25, 26, 27, 28, 30, 35, 39, 40, 42, 43, 46, 48, 49, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 16, 18, 24, 25, 26, 27, 35, 39, 40, 42, 47, 48, 49, 51, 52, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 16, 18, 24, 25, 26, 27, 28, 30, 35, 39, 40, 42, 47, 48, 49, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 26, 27, 35, 36, 38, 39, 40, 42, 43, 49, 51, 52, 56, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 26, 27, 33, 35, 38, 39, 40, 42, 46, 49, 51, 52, 54, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 26, 27, 28, 30, 35, 36, 38, 39, 40, 42, 43, 49, 56, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 18, 26, 27, 28, 30, 33, 35, 38, 39, 40, 42, 46, 49, 54, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 18, 26, 27, 35, 36, 37, 39, 40, 42, 44, 49, 51, 52, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 18, 26, 27, 34, 35, 37, 39, 40, 42, 46, 49, 51, 52, 55, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 18, 26, 27, 28, 30, 35, 36, 37, 39, 40, 42, 44, 49, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 18, 26, 27, 28, 30, 34, 35, 37, 39, 40, 42, 46, 49, 55, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 19, 26, 34, 36, 37, 40, 41, 44, 45, 46, 52, 53, 55, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 17, 18, 19, 23, 26, 36, 38, 43, 46, 49, 50, 51, 58, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 23, 26, 28, 31, 33, 38, 40, 46, 49, 50, 51, 53, 58, 59, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 18, 19, 23, 25, 28, 30, 33, 38, 40, 46, 47, 48, 50, 52, 53, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18, 19, 23, 24, 25, 27, 30, 40, 43, 46, 48, 51, 52, 53, 54, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 18, 20, 22, 23, 24, 25, 27, 29, 30, 38, 43, 45, 47, 49, 52, 53, 54, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 18, 19, 23, 26, 30, 31, 32, 33, 38, 40, 46, 50, 52, 53, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 15, 18, 20, 22, 26, 27, 29, 30, 33, 43, 44, 45, 46, 47, 49, 52, 53, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 15, 21, 25, 34, 35, 37, 40, 42, 43, 44, 47, 48, 51, 53, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 17, 20, 21, 25, 36, 38, 43, 44, 45, 46, 51, 53, 54, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 21, 23, 24, 34, 35, 36, 40, 42, 44, 47, 48, 51, 53, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 20, 27, 28, 29, 32, 34, 35, 36, 37, 42, 43, 46, 52, 54, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 18, 20, 22, 25, 26, 27, 29, 30, 43, 46, 49, 52, 54, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 18, 20, 22, 23, 25, 27, 31, 32, 43, 45, 46, 49, 54, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 17, 20, 21, 26, 34, 35, 36, 44, 46, 47, 51, 53, 56, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 18, 22, 24, 26, 27, 37, 42, 43, 45, 49, 52, 55, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 18, 22, 23, 24, 26, 27, 34, 37, 42, 43, 44, 49, 52, 53, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 18, 22, 23, 24, 27, 42, 43, 45, 47, 48, 49, 52, 53, 55, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 18, 22, 23, 24, 27, 34, 42, 45, 47, 48, 49, 52, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 20, 28, 31, 33, 35, 42, 44, 46, 49, 51, 53, 54, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 20, 21, 26, 27, 30, 34, 38, 44, 46, 47, 49, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 20, 28, 31, 33, 37, 38, 42, 44, 46, 49, 51, 54, 57, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 20, 28, 31, 33, 34, 35, 37, 42, 44, 46, 49, 51, 56, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 23, 24, 26, 27, 29, 30, 31, 33, 34, 42, 44, 48, 51, 56, 57, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 23, 24, 26, 27, 29, 30, 31, 33, 34, 41, 46, 48, 51, 56, 57, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 19, 24, 25, 26, 29, 32, 38, 41, 44, 46, 47, 49, 52, 54, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 21, 23, 24, 29, 30, 34, 35, 40, 42, 44, 47, 48, 51, 52, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 17, 20, 21, 26, 29, 30, 34, 35, 44, 46, 47, 51, 52, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 17, 20, 21, 26, 28, 29, 30, 34, 35, 43, 44, 46, 47, 49, 52, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 17, 19, 26, 27, 28, 41, 43, 44, 45, 46, 49, 51, 53, 54, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 20, 22, 23, 26, 27, 28, 30, 34, 41, 45, 46, 47, 51, 52, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 18, 22, 26, 28, 29, 30, 35, 42, 43, 44, 47, 52, 54, 58, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 18, 22, 23, 26, 27, 34, 35, 36, 37, 42, 43, 44, 46, 53, 55, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 17, 18, 19, 23, 25, 28, 31, 36, 38, 45, 46, 47, 51, 52, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 20, 22, 23, 27, 33, 34, 38, 41, 45, 46, 47, 48, 53, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 22, 29, 32, 38, 42, 43, 45, 47, 49, 52, 53, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 22, 26, 27, 38, 42, 43, 45, 46, 53, 54, 55, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 17, 20, 21, 23, 29, 30, 34, 36, 44, 48, 51, 52, 53, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 21, 23, 29, 30, 34, 35, 37, 40, 42, 44, 48, 51, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 19, 25, 29, 30, 32, 40, 41, 43, 45, 46, 48, 49, 56, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 19, 24, 29, 30, 32, 40, 41, 44, 45, 46, 47, 49, 56, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 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58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 24, 26, 30, 33, 40, 48, 49, 52, 55, 56, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 27, 29, 33, 35, 36, 38, 39, 42, 43, 44, 46, 51, 53, 54 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 20, 27, 29, 33, 34, 37, 39, 42, 44, 51, 53, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 20, 27, 29, 34, 35, 36, 37, 39, 42, 43, 44, 46, 51, 53, 55 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 18, 30, 34, 35, 36, 39, 41, 43, 45, 47, 49, 52, 54, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 18, 25, 30, 35, 39, 41, 43, 45, 47, 48, 49, 52, 54, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 20, 28, 35, 38, 39, 42, 43, 45, 47, 49, 51, 53, 54, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 30, 35, 38, 39, 41, 43, 45, 47, 49, 52, 53, 54, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 20, 27, 29, 34, 39, 42, 44, 51, 53, 54, 59, 61, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 147 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS19[1], DS19[3], DS19[6], DS19[9], DS19[12], DS19[17], DS19[19], DS19[21], DS19[22], DS19[24], DS19[25], DS19[26], DS19[27], DS19[28], DS19[29], DS19[30], DS19[31], DS19[32], DS19[33], DS19[34], DS19[35], DS19[36], DS19[37], DS19[38], DS19[39], DS19[40], DS19[41], DS19[42], DS19[43], DS19[44], DS19[45], DS19[46], DS19[47], DS19[48], DS19[49], DS19[50], DS19[51], DS19[52], DS19[53], DS19[54], DS19[55], DS19[56], DS19[57], DS19[58], DS19[59], DS19[60], DS19[61], DS19[62], DS19[63], DS19[64], DS19[65], DS19[67], DS19[68], DS19[69], DS19[70], DS19[71], DS19[72], DS19[73], DS19[74], DS19[75], DS19[76], DS19[77], DS19[78], DS19[79], DS19[80], DS19[81], DS19[82], DS19[83], DS19[85], DS19[86], DS19[87], DS19[88], DS19[89], DS19[90], DS19[91], DS19[92], DS19[93], DS19[94], DS19[95], DS19[96], DS19[97], DS19[99], DS19[100], DS19[101], DS19[102], DS19[103], DS19[104], DS19[105], DS19[106], DS19[107], DS19[108], DS19[109], DS19[111], DS19[114], DS19[115], DS19[116], DS19[117], DS19[120], DS19[121], DS19[122], DS19[123], DS19[124], DS19[125], DS19[126], DS19[127], DS19[128], DS19[129], DS19[130], DS19[131], DS19[132], DS19[133], DS19[134], DS19[135], DS19[136], DS19[137], DS19[139], DS19[140], DS19[141], DS19[142], DS19[143], DS19[144], DS19[145], DS19[146], DS19[147], DS19[149], DS19[150], DS19[151], DS19[152], DS19[153], DS19[154], DS19[155], DS19[156], DS19[157], DS19[158], DS19[159], DS19[160], DS19[161], DS19[162], DS19[163], DS19[164], DS19[166], DS19[167], DS19[172], DS19[173], DS19[174], DS19[175], DS19[176]. # Note that there is no reversible (64,28,12) difference sets in # group [64,19]. ########################### ########################### ###########################