### 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 [64,183]. 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 [64,183]. # 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(64,183));' can be # used to create a list e of the elements of the group [64,183]. # In our list of (64,28,12) DSs in group [64,183], we have [1, # 2, 3, 4, 5, 6, 8, 10, 12, 13, 17, 20, 21, 22, 26, # 30, 34, 36, 40, 43, 47, 49, 50, 55, 57, 61, 63, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[10], e[12], e[13], e[17], e[20], e[21], e[22], e[26], e[30], # e[34], e[36], e[40], e[43], e[47], e[49], e[50], e[55], e[57], e[61], e[63], e[64]] # where e is the list of the elements of the group [64,183] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,183));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 183] ### The number of inequivalent (64,28,12) DSs in group [64,183] is 160 ### The list of all inequivalent (64,28,12) DSs in group [64,183] is DS183:=[ [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 17, 20, 21, 22, 26, 30, 34, 36, 40, 43, 47, 49, 50, 55, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 17, 20, 21, 22, 26, 27, 30, 34, 36, 40, 47, 55, 57, 58, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 25, 26, 27, 28, 31, 36, 37, 39, 41, 42, 48, 50, 51, 54, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 20, 25, 26, 27, 28, 31, 37, 41, 48, 50, 51, 53, 54, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 20, 21, 26, 30, 40, 41, 42, 43, 47, 49, 50, 53, 55, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 17, 20, 21, 26, 27, 30, 40, 41, 42, 47, 53, 55, 58, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 25, 26, 27, 28, 31, 36, 38, 39, 40, 42, 48, 50, 51, 54, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 22, 26, 27, 30, 36, 39, 40, 41, 47, 54, 56, 57, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 20, 26, 27, 31, 38, 40, 46, 49, 50, 53, 54, 55, 56, 57, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 20, 25, 26, 31, 38, 40, 43, 44, 48, 53, 54, 55, 56, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 20, 25, 26, 27, 28, 31, 38, 40, 48, 50, 51, 53, 54, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 18, 20, 22, 26, 27, 30, 40, 41, 42, 47, 53, 54, 58, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 20, 22, 25, 29, 32, 34, 37, 39, 42, 43, 45, 47, 48, 50, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 18, 22, 29, 32, 37, 41, 43, 44, 45, 47, 49, 50, 54, 58, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 18, 22, 29, 30, 32, 37, 41, 43, 44, 45, 46, 47, 50, 52, 54, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 18, 22, 25, 29, 32, 37, 41, 43, 45, 46, 47, 48, 49, 50, 54, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 18, 22, 25, 29, 30, 32, 37, 41, 43, 45, 47, 48, 50, 52, 54, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 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11, 12, 13, 14, 18, 22, 26, 27, 30, 40, 41, 44, 47, 48, 51, 52, 54, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 22, 26, 27, 30, 32, 40, 41, 44, 47, 54, 58, 59, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 32, 38, 39, 40, 42, 43, 44, 47, 50, 54, 55, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 31, 38, 39, 40, 42, 43, 44, 48, 51, 54, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 28, 32, 38, 40, 42, 43, 47, 50, 51, 53, 54, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 28, 31, 32, 38, 40, 43, 51, 53, 54, 55, 56, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 27, 31, 38, 40, 42, 44, 48, 50, 51, 53, 54, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 26, 27, 31, 32, 38, 40, 44, 50, 53, 54, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 32, 39, 40, 41, 43, 44, 47, 50, 54, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 31, 39, 40, 41, 43, 44, 48, 51, 54, 56, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 28, 40, 41, 42, 43, 47, 48, 50, 53, 54, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 28, 31, 40, 41, 43, 48, 53, 54, 56, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 27, 40, 41, 42, 44, 47, 48, 51, 53, 54, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 18, 20, 22, 26, 27, 32, 40, 41, 44, 47, 53, 54, 56, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 18, 26, 29, 36, 38, 39, 40, 42, 43, 44, 54, 55, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 18, 20, 26, 29, 38, 40, 43, 44, 53, 54, 55, 56, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 26, 29, 30, 31, 33, 36, 37, 40, 41, 42, 44, 52, 53, 55, 57, 60 ], [ 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, 21, 26, 27, 29, 41, 42, 44, 46, 47, 49, 52, 55, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 20, 21, 26, 27, 29, 30, 41, 42, 44, 47, 55, 58, 59, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 26, 29, 30, 31, 34, 36, 38, 40, 41, 42, 44, 52, 53, 54, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 25, 26, 29, 31, 34, 36, 38, 40, 41, 42, 48, 49, 53, 54, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 20, 26, 29, 30, 31, 34, 38, 39, 40, 41, 44, 52, 54, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 20, 25, 26, 29, 31, 34, 38, 39, 40, 41, 48, 49, 54, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 26, 29, 30, 31, 36, 38, 39, 40, 42, 44, 52, 54, 55, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 26, 28, 29, 31, 36, 38, 39, 40, 42, 46, 51, 54, 55, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 25, 26, 29, 31, 36, 38, 39, 40, 42, 48, 49, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 20, 26, 29, 30, 31, 38, 40, 44, 52, 53, 54, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 20, 26, 28, 29, 31, 38, 40, 46, 51, 53, 54, 55, 56, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 18, 20, 25, 26, 29, 31, 38, 40, 48, 49, 53, 54, 55, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 19, 22, 23, 25, 31, 32, 36, 39, 40, 42, 44, 51, 54, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 19, 20, 22, 23, 25, 31, 32, 40, 44, 51, 53, 54, 56, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 25, 43, 46, 47, 50, 51, 52, 53, 58, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 25, 30, 43, 47, 49, 50, 51, 53, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 44, 46, 48, 50, 51, 52, 53, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 30, 44, 48, 49, 50, 51, 53, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 28, 46, 48, 49, 50, 52, 53, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 28, 30, 48, 50, 53, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 22, 23, 31, 36, 40, 42, 44, 48, 51, 53, 54, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 22, 23, 28, 31, 36, 39, 40, 42, 48, 54, 58, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 20, 22, 23, 31, 39, 40, 44, 48, 51, 54, 56, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 20, 22, 23, 28, 31, 40, 48, 53, 54, 56, 57, 58, 60, 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: ### DS183[6], DS183[8], DS183[9], DS183[13], DS183[15], DS183[16], DS183[17], DS183[19], DS183[21], DS183[22], DS183[23], DS183[24], DS183[25], DS183[26], DS183[27], DS183[28], DS183[29], DS183[30], DS183[31], DS183[32], DS183[33], DS183[34], DS183[35], DS183[36], DS183[37], DS183[38], DS183[39], DS183[40], DS183[41], DS183[42], DS183[43], DS183[44], DS183[45], DS183[47], DS183[48], DS183[53], DS183[54], DS183[55], DS183[57], DS183[58], DS183[59], DS183[60], DS183[61], DS183[62], DS183[63], DS183[64], DS183[65], DS183[66], DS183[67], DS183[68], DS183[70], DS183[71], DS183[72], DS183[73], DS183[74], DS183[75], DS183[76], DS183[77], DS183[78], DS183[79], DS183[81], DS183[82], DS183[83], DS183[84], DS183[85], DS183[87], DS183[89], DS183[90], DS183[91], DS183[92], DS183[93], DS183[94], DS183[95], DS183[96], DS183[97], DS183[99], DS183[100], DS183[101], DS183[102], DS183[103], DS183[104], DS183[105], DS183[106], DS183[107], DS183[108], DS183[109], DS183[110], DS183[112], DS183[113], DS183[114], DS183[115], DS183[116], DS183[117], DS183[118], DS183[119], DS183[120], DS183[121], DS183[122], DS183[123], DS183[124], DS183[125], DS183[127], DS183[128], DS183[129], DS183[130], DS183[132], DS183[133], DS183[137], DS183[138], DS183[146], DS183[147], DS183[148], DS183[149], DS183[150], DS183[154], DS183[156], DS183[157], DS183[158], DS183[159], DS183[160]. # Note that there is no reversible (64,28,12) difference sets in # group [64,183]. ########################### ########################### ###########################