### 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 [64,124]. 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 [64,124]. # 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(64,124));' can be # used to create a list e of the elements of the group [64,124]. # In our list of (64,28,12) DSs in group [64,124], we have [1, # 2, 3, 4, 5, 6, 10, 11, 13, 14, 16, 18, 19, 24, 25, # 34, 37, 41, 42, 43, 45, 47, 48, 52, 53, 54, 60, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[10], e[11], e[13], e[14], e[16], e[18], e[19], e[24], e[25], e[34], # e[37], e[41], e[42], e[43], e[45], e[47], e[48], e[52], e[53], e[54], e[60], e[64]] # where e is the list of the elements of the group [64,124] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,124));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 124] ### The number of inequivalent (64,28,12) DSs in group [64,124] is 1184 ### The list of all inequivalent (64,28,12) DSs in group [64,124] is DS124:=[ [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 14, 16, 18, 19, 24, 25, 34, 37, 41, 42, 43, 45, 47, 48, 52, 53, 54, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 14, 16, 18, 19, 24, 25, 33, 37, 41, 42, 44, 45, 47, 48, 52, 53, 55, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 15, 18, 19, 25, 26, 29, 31, 32, 34, 36, 37, 41, 42, 47, 49, 52, 53, 55, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 15, 17, 19, 25, 26, 29, 31, 32, 33, 36, 37, 40, 42, 47, 49, 52, 53, 54, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 14, 15, 17, 20, 27, 29, 30, 34, 35, 36, 41, 43, 50, 52, 53, 54, 57, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 14, 15, 17, 20, 24, 27, 29, 30, 34, 35, 36, 41, 46, 47, 50, 52, 53, 54, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 14, 15, 17, 20, 23, 25, 29, 34, 35, 36, 41, 43, 45, 48, 53, 54, 57, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 14, 15, 17, 20, 23, 24, 25, 29, 34, 35, 36, 41, 45, 46, 47, 48, 53, 54, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 15, 17, 20, 25, 29, 32, 33, 34, 35, 37, 41, 46, 48, 49, 56, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 14, 15, 17, 20, 24, 25, 27, 29, 32, 34, 35, 36, 41, 47, 48, 50, 53, 54, 57 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 14, 15, 17, 20, 23, 29, 32, 34, 35, 36, 41, 43, 45, 49, 53, 54, 57, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 14, 15, 17, 20, 23, 27, 29, 32, 34, 35, 36, 41, 45, 50, 53, 54, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 14, 15, 17, 18, 20, 29, 32, 35, 36, 39, 44, 49, 54, 55, 58, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 14, 15, 17, 18, 20, 25, 29, 32, 35, 36, 39, 46, 48, 49, 54, 55, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 17, 19, 23, 28, 32, 35, 37, 38, 40, 42, 44, 49, 50, 53, 56, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 17, 19, 20, 28, 32, 35, 36, 37, 38, 40, 43, 44, 45, 50, 53, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 17, 19, 20, 27, 28, 32, 35, 36, 37, 38, 40, 44, 45, 49, 53, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 17, 19, 20, 23, 28, 32, 35, 36, 37, 38, 40, 43, 50, 53, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 17, 19, 20, 23, 27, 28, 32, 35, 36, 37, 38, 40, 49, 53, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 19, 27, 28, 32, 35, 36, 38, 40, 42, 44, 45, 53, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 19, 20, 28, 32, 35, 38, 40, 43, 44, 45, 50, 53, 56, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 19, 20, 27, 28, 32, 35, 38, 40, 43, 44, 45, 49, 53, 56, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 19, 20, 23, 28, 32, 35, 38, 40, 43, 50, 53, 56, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 19, 20, 23, 27, 28, 32, 35, 38, 40, 43, 49, 53, 56, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 14, 18, 19, 20, 23, 24, 25, 28, 31, 38, 41, 43, 47, 48, 50, 56, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 14, 18, 19, 20, 23, 24, 25, 27, 28, 31, 38, 41, 43, 47, 48, 49, 56, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 19, 20, 28, 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38, 39, 43, 44, 45, 49, 55, 56, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 19, 20, 23, 28, 31, 34, 35, 38, 39, 43, 50, 55, 56, 57, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 19, 20, 23, 27, 28, 31, 34, 35, 38, 39, 43, 49, 55, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 28, 31, 35, 38, 41, 43, 44, 45, 50, 53, 56, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 27, 28, 31, 35, 38, 41, 43, 44, 45, 49, 53, 56, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 23, 28, 31, 35, 38, 41, 43, 50, 53, 56, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 12, 13, 14, 18, 19, 20, 23, 27, 28, 31, 35, 38, 41, 43, 49, 53, 56, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 11, 13, 16, 17, 19, 28, 29, 34, 36, 40, 42, 43, 45, 51, 52, 53, 54, 56, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 10, 11, 13, 16, 17, 19, 28, 29, 33, 36, 40, 42, 44, 45, 51, 52, 53, 55, 56, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 16, 17, 20, 25, 26, 27, 28, 29, 30, 31, 32, 34, 41, 43, 46, 48, 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46, 47, 48, 49, 50, 53, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 19, 24, 25, 26, 27, 35, 41, 45, 46, 49, 50, 56, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 19, 23, 27, 35, 36, 41, 45, 46, 49, 50, 53, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 20, 23, 25, 28, 29, 33, 39, 42, 46, 47, 50, 54, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 30, 34, 37, 38, 40, 43, 49, 52, 53, 54, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 27, 34, 36, 37, 38, 40, 43, 49, 50, 54, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 23, 25, 28, 29, 36, 39, 40, 46, 47, 50, 56, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 19, 30, 33, 37, 38, 41, 43, 49, 52, 53, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 19, 27, 33, 36, 37, 38, 41, 43, 49, 50, 55, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 19, 23, 24, 27, 36, 41, 46, 48, 49, 50, 56, 59, 60, 64 ], [ 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12, 14, 15, 17, 23, 28, 29, 36, 37, 38, 39, 40, 46, 50, 54, 55, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 19, 27, 35, 36, 37, 38, 41, 46, 49, 50, 53, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 23, 28, 29, 35, 36, 39, 40, 50, 56, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 30, 34, 41, 43, 49, 52, 53, 54, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 30, 33, 41, 44, 49, 52, 53, 55, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 27, 34, 36, 41, 43, 49, 50, 54, 56, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 27, 33, 36, 41, 44, 49, 50, 55, 56, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 23, 29, 31, 32, 33, 37, 38, 39, 40, 53, 55, 56, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 23, 28, 29, 35, 36, 39, 40, 46, 50, 53, 54, 55, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 27, 29, 36, 37, 38, 39, 40, 45, 51, 53, 56, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 23, 28, 29, 36, 37, 38, 39, 40, 50, 53, 56, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 20, 21, 23, 28, 29, 33, 38, 46, 50, 53, 56, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 19, 27, 36, 37, 40, 41, 45, 46, 50, 53, 56, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 23, 29, 31, 32, 36, 38, 39, 40, 54, 55, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 23, 28, 29, 36, 38, 39, 40, 46, 50, 56, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 29, 36, 39, 40, 45, 50, 53, 56, 61, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 890 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS124[2], DS124[17], DS124[18], DS124[20], DS124[21], DS124[30], DS124[31], DS124[34], DS124[39], DS124[43], DS124[48], DS124[49], DS124[50], DS124[53], DS124[62], DS124[63], DS124[65], DS124[71], DS124[72], DS124[80], DS124[91], DS124[92], DS124[98], DS124[102], DS124[104], DS124[105], DS124[107], DS124[108], DS124[110], DS124[111], DS124[116], DS124[117], DS124[118], DS124[119], DS124[120], DS124[124], DS124[129], DS124[132], DS124[133], DS124[134], DS124[139], DS124[141], DS124[143], DS124[146], DS124[147], DS124[149], DS124[152], DS124[153], DS124[154], DS124[158], DS124[160], DS124[164], DS124[171], DS124[179], DS124[181], DS124[182], DS124[184], DS124[185], DS124[186], DS124[188], DS124[189], DS124[190], DS124[191], DS124[192], DS124[193], DS124[194], DS124[197], DS124[198], DS124[199], DS124[200], DS124[203], DS124[205], DS124[206], DS124[210], DS124[213], DS124[216], DS124[221], DS124[222], DS124[224], DS124[225], DS124[228], DS124[235], DS124[237], DS124[238], DS124[239], DS124[240], DS124[241], DS124[242], DS124[243], DS124[252], DS124[253], DS124[255], DS124[256], DS124[262], DS124[265], DS124[266], DS124[267], DS124[270], DS124[271], DS124[272], DS124[273], DS124[274], DS124[275], DS124[277], DS124[278], DS124[279], DS124[283], DS124[285], DS124[287], DS124[288], DS124[291], DS124[294], DS124[295], DS124[296], DS124[298], DS124[299], DS124[300], DS124[305], DS124[306], DS124[307], DS124[308], DS124[309], DS124[310], DS124[311], DS124[313], DS124[314], DS124[315], DS124[317], DS124[318], DS124[322], DS124[323], DS124[324], DS124[325], DS124[326], DS124[330], DS124[333], DS124[335], DS124[336], DS124[337], DS124[340], DS124[341], DS124[343], DS124[344], DS124[345], DS124[346], DS124[347], DS124[348], DS124[349], DS124[350], DS124[353], DS124[354], DS124[355], DS124[356], DS124[357], DS124[359], DS124[361], DS124[362], DS124[363], DS124[364], DS124[365], DS124[366], DS124[367], DS124[368], DS124[374], DS124[376], DS124[378], DS124[382], DS124[385], DS124[387], DS124[389], DS124[390], DS124[394], DS124[396], DS124[397], DS124[398], DS124[399], DS124[400], DS124[403], DS124[405], DS124[406], DS124[407], DS124[408], DS124[409], DS124[410], DS124[411], DS124[412], DS124[413], DS124[414], DS124[415], DS124[416], DS124[417], DS124[418], DS124[419], DS124[420], DS124[421], DS124[422], DS124[423], DS124[424], DS124[425], DS124[429], DS124[430], DS124[431], DS124[432], DS124[433], DS124[434], DS124[435], DS124[436], DS124[437], DS124[438], DS124[440], DS124[441], DS124[442], DS124[444], DS124[445], DS124[446], DS124[447], DS124[448], DS124[449], DS124[451], DS124[453], DS124[457], DS124[458], DS124[459], DS124[463], DS124[464], DS124[465], DS124[466], DS124[467], DS124[468], DS124[469], DS124[470], DS124[471], DS124[472], DS124[473], DS124[474], DS124[475], DS124[476], DS124[481], DS124[492], DS124[493], DS124[498], DS124[499], DS124[505], DS124[507], DS124[511], DS124[512], DS124[513], DS124[514], DS124[515], DS124[516], DS124[517], DS124[518], DS124[519], DS124[520], DS124[521], DS124[522], DS124[523], DS124[524], DS124[525], DS124[526], DS124[527], 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DS124[1128], DS124[1129], DS124[1130], DS124[1131], DS124[1132], DS124[1133], DS124[1134], DS124[1135], DS124[1136], DS124[1137], DS124[1138], DS124[1139], DS124[1140], DS124[1141], DS124[1142], DS124[1143], DS124[1144], DS124[1145], DS124[1146], DS124[1147], DS124[1148], DS124[1149], DS124[1150], DS124[1151], DS124[1152], DS124[1153], DS124[1154], DS124[1155], DS124[1156], DS124[1157], DS124[1158], DS124[1159], DS124[1160], DS124[1161], DS124[1162], DS124[1163], DS124[1164], DS124[1165], DS124[1166], DS124[1167], DS124[1168], DS124[1169], DS124[1170], DS124[1171], DS124[1172], DS124[1173], DS124[1174], DS124[1175], DS124[1176], DS124[1177], DS124[1178], DS124[1179], DS124[1180], DS124[1181], DS124[1182], DS124[1183], DS124[1184]. # Note that there is no reversible (64,28,12) difference sets in # group [64,124]. ########################### ########################### ###########################