### 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 [64,14]. 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 [64,14]. # 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(64,14));' can be # used to create a list e of the elements of the group [64,14]. # In our list of (64,28,12) DSs in group [64,14], we have [1, # 2, 3, 4, 5, 6, 8, 13, 16, 17, 18, 20, 23, 25, 26, # 30, 31, 32, 33, 35, 36, 39, 43, 48, 54, 56, 60, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[13], e[16], e[17], e[18], e[20], e[23], e[25], e[26], e[30], # e[31], e[32], e[33], e[35], e[36], e[39], e[43], e[48], e[54], e[56], e[60], e[64]] # where e is the list of the elements of the group [64,14] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,14));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 14] ### The number of inequivalent (64,28,12) DSs in group [64,14] is 319 ### The list of all inequivalent (64,28,12) DSs in group [64,14] is DS14:=[ [ 1, 2, 3, 4, 5, 6, 8, 13, 16, 17, 18, 20, 23, 25, 26, 30, 31, 32, 33, 35, 36, 39, 43, 48, 54, 56, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 16, 17, 18, 20, 23, 24, 30, 31, 32, 33, 35, 36, 39, 44, 45, 48, 54, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 17, 18, 20, 23, 26, 30, 31, 32, 34, 35, 38, 39, 43, 45, 46, 55, 56, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 17, 18, 20, 23, 24, 26, 30, 31, 32, 34, 35, 38, 39, 45, 47, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 16, 17, 18, 20, 23, 25, 26, 30, 31, 32, 35, 37, 39, 43, 48, 53, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 13, 14, 16, 17, 18, 20, 23, 24, 30, 31, 32, 35, 37, 39, 44, 45, 48, 53, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 12, 17, 18, 20, 23, 24, 28, 30, 36, 37, 38, 39, 44, 45, 48, 49, 51, 52, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 18, 20, 24, 26, 32, 38, 39, 43, 44, 47, 53, 54, 55, 56, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 18, 20, 23, 26, 32, 38, 39, 43, 45, 46, 53, 54, 55, 56, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 18, 20, 23, 24, 26, 32, 38, 39, 45, 47, 53, 54, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 14, 16, 17, 18, 20, 25, 26, 27, 28, 29, 32, 35, 37, 39, 48, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 17, 18, 20, 23, 24, 27, 28, 31, 36, 37, 38, 39, 44, 45, 48, 50, 51, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 23, 24, 27, 28, 31, 38, 39, 48, 50, 51, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 17, 18, 20, 26, 29, 34, 35, 36, 37, 38, 39, 43, 44, 53, 54, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 17, 18, 20, 23, 24, 27, 29, 30, 36, 37, 38, 39, 44, 45, 48, 50, 52, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 16, 17, 18, 20, 23, 24, 27, 29, 30, 38, 39, 48, 50, 52, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 18, 26, 27, 29, 33, 34, 35, 36, 38, 42, 44, 49, 50, 53, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 18, 20, 26, 29, 33, 34, 35, 38, 39, 43, 44, 56, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 18, 20, 25, 26, 27, 29, 30, 33, 34, 35, 38, 39, 48, 50, 52, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 18, 24, 25, 26, 29, 35, 39, 40, 42, 47, 48, 53, 54, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 17, 18, 20, 24, 26, 29, 33, 36, 37, 38, 39, 44, 46, 47, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 17, 18, 20, 24, 25, 26, 29, 33, 36, 37, 38, 39, 47, 48, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 18, 20, 27, 29, 30, 33, 38, 39, 46, 47, 48, 50, 52, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 17, 18, 26, 29, 33, 36, 38, 42, 43, 44, 53, 55, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 17, 18, 20, 27, 29, 30, 34, 35, 38, 39, 46, 47, 48, 50, 52, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 17, 18, 20, 25, 26, 27, 29, 30, 38, 39, 48, 50, 52, 53, 54, 55, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 17, 18, 20, 24, 26, 29, 38, 39, 44, 46, 47, 53, 54, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 17, 18, 20, 24, 25, 26, 29, 38, 39, 47, 48, 53, 54, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 18, 26, 27, 29, 34, 39, 40, 42, 44, 49, 50, 53, 54, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 18, 24, 26, 29, 34, 39, 40, 42, 44, 46, 47, 53, 54, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 18, 24, 26, 29, 33, 34, 39, 40, 42, 44, 46, 47, 56, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 17, 18, 26, 27, 29, 34, 42, 44, 49, 50, 53, 54, 56, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 23, 24, 26, 28, 31, 32, 38, 40, 41, 42, 45, 47, 52, 53, 54, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 23, 24, 26, 28, 31, 32, 33, 38, 40, 41, 42, 45, 47, 52, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 15, 17, 18, 21, 27, 32, 33, 35, 36, 40, 46, 47, 48, 51, 54, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 20, 23, 26, 27, 32, 36, 39, 40, 41, 43, 45, 46, 51, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 21, 27, 32, 35, 37, 40, 46, 47, 48, 51, 53, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 16, 18, 21, 27, 32, 33, 35, 37, 39, 46, 47, 48, 51, 53, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 18, 21, 27, 32, 35, 36, 39, 46, 47, 48, 51, 54, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 18, 21, 27, 32, 33, 34, 37, 39, 46, 47, 48, 51, 54, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 17, 18, 21, 27, 32, 34, 37, 40, 46, 47, 48, 51, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 17, 18, 20, 23, 26, 28, 29, 32, 36, 37, 38, 39, 43, 45, 46, 51, 52, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 17, 18, 20, 23, 24, 26, 28, 29, 32, 36, 37, 38, 39, 45, 47, 51, 52, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 17, 18, 20, 25, 26, 32, 34, 36, 39, 48, 49, 50, 51, 52, 53, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 21, 26, 32, 35, 42, 44, 46, 49, 50, 51, 52, 55, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 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49, 53, 54, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 26, 27, 28, 30, 33, 38, 41, 42, 43, 44, 47, 49, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 25, 28, 31, 38, 41, 42, 43, 45, 47, 49, 53, 54, 59, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 24, 25, 28, 31, 33, 38, 41, 42, 43, 45, 47, 49, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 26, 28, 31, 38, 41, 42, 45, 47, 49, 53, 54, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 26, 28, 31, 33, 38, 41, 42, 45, 47, 49, 59, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 26, 27, 28, 30, 38, 41, 42, 43, 45, 46, 49, 53, 54, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 26, 27, 28, 30, 33, 38, 41, 42, 43, 45, 46, 49, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 25, 28, 31, 38, 41, 42, 44, 46, 49, 53, 54, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 25, 28, 31, 33, 38, 41, 42, 44, 46, 49, 58, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 25, 27, 28, 31, 38, 41, 42, 43, 44, 46, 50, 53, 54, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 26, 28, 31, 38, 41, 42, 45, 46, 49, 53, 54, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 26, 28, 31, 38, 41, 42, 43, 45, 46, 49, 53, 54, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 26, 28, 31, 33, 38, 41, 42, 43, 45, 46, 49, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 26, 27, 28, 30, 38, 41, 42, 45, 47, 49, 53, 54, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 26, 27, 28, 30, 33, 38, 41, 42, 45, 47, 49, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 25, 28, 31, 38, 41, 42, 43, 44, 47, 49, 53, 54, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 25, 28, 31, 33, 38, 41, 42, 43, 44, 47, 49, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 25, 27, 28, 31, 38, 41, 42, 44, 47, 50, 53, 54, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 22, 23, 24, 25, 27, 28, 31, 33, 38, 41, 42, 44, 47, 50, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 20, 22, 27, 33, 35, 36, 48, 49, 51, 52, 54, 56, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 20, 22, 27, 33, 35, 36, 43, 48, 49, 51, 52, 54, 56, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 27, 35, 37, 48, 49, 51, 52, 53, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 27, 35, 37, 43, 48, 49, 51, 52, 53, 63, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 120 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS14[10], DS14[12], DS14[15], DS14[20], DS14[22], DS14[27], DS14[29], DS14[30], DS14[31], DS14[47], DS14[50], DS14[54], DS14[55], DS14[57], DS14[60], DS14[62], DS14[63], DS14[69], DS14[70], DS14[72], DS14[75], DS14[76], DS14[77], DS14[78], DS14[79], DS14[81], DS14[82], DS14[83], DS14[84], DS14[88], DS14[90], DS14[94], DS14[95], DS14[97], DS14[98], DS14[99], DS14[100], DS14[102], DS14[103], DS14[105], DS14[106], DS14[109], DS14[113], DS14[121], DS14[126], DS14[129], DS14[134], DS14[142], DS14[145], DS14[153], DS14[166], DS14[167], DS14[168], DS14[169], DS14[170], DS14[171], DS14[172], DS14[173], DS14[175], DS14[177], DS14[180], DS14[181], DS14[182], DS14[183], DS14[184], DS14[186], DS14[190], DS14[191], DS14[192], DS14[193], DS14[202], DS14[203], DS14[204], DS14[205], DS14[207], DS14[210], DS14[211], DS14[212], DS14[213], DS14[214], DS14[217], DS14[219], DS14[220], DS14[223], DS14[225], DS14[226], DS14[228], DS14[229], DS14[231], DS14[239], DS14[241], DS14[243], DS14[249], DS14[250], DS14[252], DS14[262], DS14[263], DS14[265], DS14[267], DS14[268], DS14[269], DS14[272], DS14[275], DS14[279], DS14[283], DS14[288], DS14[289], DS14[290], DS14[296], DS14[299], DS14[300], DS14[301], DS14[306], DS14[308], DS14[309], DS14[311], DS14[312], DS14[317], DS14[318], DS14[319]. # Note that there is no reversible (64,28,12) difference sets in # group [64,14]. ########################### ########################### ###########################