### 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 [64,147]. Also, we determine the nonisomorphic # symmetric (64,28,12) designs that rise from these DSs. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,147]. # 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(64,147));' can be # used to create a list e of the elements of the group [64,147]. # In our list of (64,28,12) DSs in group [64,147], we have [1, # 2, 3, 4, 5, 7, 9, 10, 11, 14, 19, 21, 23, 28, 30, # 33, 34, 35, 38, 39, 44, 45, 50, 56, 57, 60, 62, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[7], e[9], e[10], e[11], e[14], e[19], e[21], e[23], e[28], e[30], e[33], # e[34], e[35], e[38], e[39], e[44], e[45], e[50], e[56], e[57], e[60], e[62], e[63]] # where e is the list of the elements of the group [64,147] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,147));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 147] ### The number of inequivalent (64,28,12) DSs in group [64,147] is 236 ### The list of all inequivalent (64,28,12) DSs in group [64,147] is DS147:=[ [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 14, 19, 21, 23, 28, 30, 33, 34, 35, 38, 39, 44, 45, 50, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 14, 19, 21, 23, 28, 30, 38, 39, 44, 45, 50, 53, 54, 55, 56, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 14, 19, 21, 25, 26, 27, 28, 31, 33, 34, 35, 38, 39, 48, 49, 52, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 19, 25, 26, 27, 28, 31, 37, 38, 39, 40, 42, 48, 49, 52, 53, 55, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 19, 21, 25, 26, 27, 28, 31, 38, 39, 48, 49, 52, 53, 54, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 23, 27, 30, 34, 35, 38, 40, 43, 45, 51, 52, 55, 56, 57, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 23, 27, 29, 30, 34, 35, 38, 40, 43, 50, 52, 55, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 16, 18, 20, 23, 32, 34, 35, 36, 40, 43, 45, 49, 50, 51, 52, 55, 56, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 16, 18, 20, 23, 32, 34, 35, 37, 40, 45, 49, 50, 51, 52, 55, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 20, 23, 29, 31, 34, 35, 38, 40, 43, 49, 55, 56, 57, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 20, 23, 27, 31, 34, 35, 38, 40, 45, 51, 55, 56, 57, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 16, 18, 20, 25, 26, 27, 30, 31, 32, 34, 36, 40, 48, 51, 53, 55, 56, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 16, 18, 20, 21, 25, 27, 30, 31, 32, 34, 36, 42, 45, 51, 53, 55, 56, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 16, 18, 20, 21, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 42, 45, 50, 56, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 16, 18, 20, 25, 26, 27, 30, 31, 32, 33, 34, 37, 40, 48, 51, 54, 55, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 16, 18, 20, 21, 25, 29, 30, 31, 32, 33, 34, 35, 37, 42, 45, 49, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 16, 18, 20, 21, 25, 27, 30, 31, 32, 33, 34, 37, 42, 45, 51, 54, 55, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 18, 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9, 10, 11, 16, 19, 21, 22, 23, 28, 30, 43, 46, 47, 48, 50, 53, 60, 62, 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, 19, 22, 26, 28, 31, 34, 36, 38, 42, 49, 50, 54, 55, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 19, 22, 26, 27, 28, 31, 34, 36, 38, 42, 43, 54, 55, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 21, 22, 28, 33, 35, 37, 41, 43, 48, 49, 52, 55, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 21, 22, 27, 28, 33, 35, 37, 41, 48, 50, 52, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 19, 22, 26, 28, 31, 35, 42, 43, 49, 50, 53, 55, 56, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 19, 22, 26, 27, 28, 31, 35, 42, 53, 55, 56, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 22, 28, 30, 33, 35, 36, 41, 42, 43, 48, 50, 54, 56, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 19, 21, 22, 28, 30, 36, 37, 48, 50, 53, 55, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 22, 28, 30, 35, 37, 41, 42, 43, 48, 50, 53, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 21, 22, 28, 36, 41, 43, 48, 49, 52, 54, 55, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 21, 22, 27, 28, 36, 41, 48, 50, 52, 54, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 19, 21, 22, 28, 30, 43, 48, 50, 53, 55, 56, 62, 63, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 124 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS147[3], DS147[4], DS147[11], DS147[15], DS147[23], DS147[25], DS147[35], DS147[37], DS147[39], DS147[40], DS147[41], DS147[43], DS147[46], DS147[50], DS147[51], DS147[55], DS147[56], DS147[57], DS147[58], DS147[59], DS147[60], DS147[61], DS147[62], DS147[63], DS147[64], DS147[66], DS147[67], DS147[68], DS147[69], DS147[70], DS147[71], DS147[73], DS147[78], DS147[79], DS147[80], DS147[81], DS147[90], DS147[91], DS147[92], DS147[93], DS147[94], DS147[99], DS147[100], DS147[104], DS147[105], DS147[108], DS147[109], DS147[110], DS147[111], DS147[112], DS147[114], DS147[115], DS147[121], DS147[125], DS147[128], DS147[131], DS147[133], DS147[134], DS147[140], DS147[146], DS147[148], DS147[149], DS147[150], DS147[153], DS147[154], DS147[155], DS147[156], DS147[157], DS147[158], DS147[159], DS147[160], DS147[161], DS147[162], DS147[163], DS147[164], DS147[165], DS147[166], DS147[168], DS147[169], DS147[170], DS147[171], DS147[172], DS147[173], DS147[174], DS147[175], DS147[176], DS147[178], DS147[179], DS147[180], DS147[181], DS147[182], DS147[184], DS147[185], DS147[186], DS147[187], DS147[188], DS147[189], DS147[190], DS147[192], DS147[196], DS147[199], DS147[202], DS147[203], DS147[206], DS147[207], DS147[210], DS147[213], DS147[214], DS147[215], DS147[216], DS147[219], DS147[220], DS147[221], DS147[223], DS147[224], DS147[225], DS147[226], DS147[227], DS147[228], DS147[229], DS147[230], DS147[234], DS147[235], DS147[236]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,147]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,147] is 20 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,147] is RDS147:=[ [ 2, 4, 6, 8, 9, 10, 11, 13, 19, 20, 22, 24, 28, 30, 33, 39, 42, 43, 48, 50, 55, 57, 58, 59, 61, 62, 63, 64 ], [ 2, 4, 6, 8, 9, 10, 11, 13, 19, 20, 22, 23, 24, 28, 30, 33, 39, 42, 44, 45, 48, 50, 55, 57, 60, 61, 62, 63 ], [ 2, 4, 5, 6, 8, 9, 10, 11, 13, 17, 19, 21, 22, 24, 28, 30, 33, 40, 43, 48, 50, 55, 58, 59, 61, 62, 63, 64 ], [ 2, 4, 5, 6, 8, 9, 10, 11, 13, 17, 19, 21, 22, 23, 24, 28, 30, 33, 40, 44, 45, 48, 50, 55, 60, 61, 62, 63 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 16, 21, 22, 24, 28, 30, 33, 37, 43, 48, 50, 55, 58, 59, 61, 62, 63, 64 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 16, 21, 22, 23, 24, 28, 30, 33, 37, 44, 45, 48, 50, 55, 60, 61, 62, 63 ], [ 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 21, 22, 24, 28, 30, 33, 37, 39, 40, 48, 50, 55, 57, 61, 62, 63 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 14, 16, 19, 24, 28, 30, 33, 37, 39, 43, 48, 50, 55, 57, 58, 59, 61, 62, 63, 64 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 14, 16, 19, 23, 24, 28, 30, 33, 37, 39, 44, 45, 48, 50, 55, 57, 60, 61, 62, 63 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 16, 19, 20, 21, 24, 28, 30, 33, 37, 39, 42, 48, 50, 55, 57, 61, 62, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 18, 19, 21, 22, 24, 28, 30, 33, 37, 41, 48, 50, 55, 61, 62, 63 ], [ 1, 2, 4, 7, 8, 9, 10, 11, 13, 17, 19, 20, 24, 28, 30, 33, 40, 42, 43, 48, 50, 55, 58, 59, 61, 62, 63, 64 ], [ 1, 2, 4, 7, 8, 9, 10, 11, 13, 17, 19, 20, 23, 24, 28, 30, 33, 40, 42, 44, 45, 48, 50, 55, 60, 61, 62, 63 ], [ 1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 19, 21, 24, 28, 30, 33, 39, 43, 48, 50, 55, 57, 58, 59, 61, 62, 63, 64 ], [ 1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 19, 21, 23, 24, 28, 30, 33, 39, 44, 45, 48, 50, 55, 57, 60, 61, 62, 63 ], [ 1, 2, 3, 7, 8, 9, 10, 11, 13, 14, 16, 20, 24, 28, 30, 33, 37, 42, 43, 48, 50, 55, 58, 59, 61, 62, 63, 64 ], [ 1, 2, 3, 7, 8, 9, 10, 11, 13, 14, 16, 20, 23, 24, 28, 30, 33, 37, 42, 44, 45, 48, 50, 55, 60, 61, 62, 63 ], [ 1, 2, 3, 7, 8, 9, 10, 11, 13, 14, 16, 17, 20, 24, 28, 30, 33, 37, 39, 40, 42, 48, 50, 55, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 13, 14, 16, 18, 19, 20, 24, 28, 30, 33, 37, 41, 42, 48, 50, 55, 61, 62, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 16, 19, 22, 24, 28, 30, 33, 37, 39, 48, 50, 55, 57, 61, 62, 63 ] ]; ########################### ########################### ###########################