### 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 [64,175]. 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 [64,175]. # 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(64,175));' can be # used to create a list e of the elements of the group [64,175]. # In our list of (64,28,12) DSs in group [64,175], we have [1, # 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 20, 24, 25, # 27, 29, 30, 33, 41, 50, 52, 53, 54, 56, 57, 61, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[9], e[12], e[14], e[15], e[17], e[20], e[24], e[25], e[27], # e[29], e[30], e[33], e[41], e[50], e[52], e[53], e[54], e[56], e[57], e[61], e[63]] # where e is the list of the elements of the group [64,175] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,175));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 175] ### The number of inequivalent (64,28,12) DSs in group [64,175] is 623 ### The list of all inequivalent (64,28,12) DSs in group [64,175] is DS175:=[ [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 17, 20, 24, 25, 27, 29, 30, 33, 41, 50, 52, 53, 54, 56, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 18, 20, 25, 26, 27, 29, 30, 33, 38, 40, 48, 50, 52, 55, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 18, 19, 20, 24, 25, 27, 29, 30, 33, 38, 41, 46, 50, 52, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 13, 14, 15, 17, 20, 24, 25, 27, 29, 30, 34, 35, 41, 50, 52, 55, 56, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 19, 20, 25, 26, 27, 29, 31, 33, 38, 39, 48, 50, 52, 55, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 17, 19, 21, 25, 26, 27, 31, 37, 38, 39, 48, 51, 52, 53, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 17, 19, 20, 25, 26, 27, 31, 35, 38, 40, 48, 51, 52, 53, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 20, 25, 26, 27, 31, 35, 37, 39, 40, 41, 48, 51, 52, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 19, 20, 25, 26, 27, 31, 33, 37, 39, 48, 51, 52, 55, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 20, 25, 26, 27, 31, 35, 37, 41, 48, 51, 52, 53, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 18, 20, 27, 29, 32, 35, 38, 40, 46, 47, 48, 50, 53, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 18, 20, 26, 27, 29, 32, 33, 38, 40, 46, 50, 55, 56, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 18, 20, 24, 27, 29, 32, 35, 38, 40, 48, 50, 53, 56, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 17, 20, 24, 25, 27, 29, 32, 35, 41, 50, 53, 56, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 17, 19, 23, 26, 28, 29, 30, 32, 38, 40, 42, 43, 44, 46, 51, 56, 58, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 17, 19, 20, 23, 26, 28, 32, 38, 40, 44, 45, 46, 50, 52, 56, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 17, 19, 25, 28, 29, 30, 32, 40, 42, 43, 44, 46, 47, 51, 56, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 17, 19, 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12, 13, 14, 25, 30, 37, 38, 39, 40, 44, 50, 52, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 29, 33, 37, 38, 39, 40, 46, 49, 51, 53, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 28, 30, 31, 35, 41, 46, 48, 49, 50, 53, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 25, 28, 29, 37, 38, 39, 40, 46, 54, 55, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 24, 25, 27, 30, 32, 35, 41, 44, 50, 53, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 26, 36, 38, 40, 42, 45, 49, 50, 53, 55, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 24, 27, 28, 29, 35, 36, 42, 44, 55, 56, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 23, 28, 35, 36, 42, 43, 47, 49, 51, 55, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 25, 37, 38, 39, 41, 46, 49, 50, 54, 55, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 25, 27, 33, 34, 37, 38, 41, 46, 49, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 25, 30, 37, 38, 40, 41, 50, 52, 54, 55, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 25, 29, 30, 31, 43, 46, 48, 49, 54, 55, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 25, 29, 33, 34, 37, 38, 39, 46, 51, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 37, 39, 40, 44, 48, 50, 52, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 37, 39, 40, 46, 48, 49, 51, 54, 55, 57, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 32, 37, 39, 40, 46, 52, 54, 55, 57, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 32, 37, 39, 40, 44, 46, 52, 54, 55, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 28, 29, 37, 39, 40, 46, 48, 54, 55, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 33, 37, 39, 41, 46, 48, 49, 50, 53, 55, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 30, 37, 40, 41, 48, 50, 52, 54, 55, 58, 63, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 475 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS175[4], DS175[5], DS175[6], DS175[7], DS175[8], DS175[9], DS175[11], DS175[12], DS175[13], DS175[14], DS175[16], DS175[17], DS175[18], DS175[19], DS175[20], DS175[23], DS175[24], DS175[25], DS175[27], DS175[28], DS175[29], DS175[30], DS175[32], DS175[34], DS175[35], DS175[36], DS175[37], DS175[41], DS175[44], DS175[45], DS175[46], DS175[47], DS175[48], DS175[49], DS175[50], DS175[51], DS175[52], DS175[53], DS175[54], DS175[55], DS175[57], DS175[58], DS175[59], DS175[60], DS175[62], DS175[63], DS175[64], DS175[65], DS175[68], DS175[72], DS175[73], DS175[77], DS175[78], DS175[79], DS175[80], DS175[84], DS175[88], DS175[89], DS175[92], DS175[93], DS175[95], DS175[96], DS175[97], DS175[99], DS175[101], DS175[102], DS175[103], DS175[105], DS175[106], DS175[108], DS175[110], DS175[112], DS175[113], DS175[114], DS175[115], DS175[116], DS175[118], DS175[119], DS175[120], DS175[121], DS175[126], DS175[127], DS175[129], DS175[131], DS175[132], DS175[136], DS175[138], DS175[139], DS175[142], DS175[143], DS175[144], DS175[146], DS175[147], DS175[149], DS175[150], DS175[151], DS175[152], DS175[153], DS175[154], DS175[157], DS175[158], DS175[159], DS175[160], DS175[162], DS175[163], DS175[164], DS175[165], DS175[166], DS175[167], DS175[169], DS175[170], DS175[171], DS175[172], DS175[174], DS175[175], DS175[176], DS175[177], DS175[178], DS175[179], DS175[181], DS175[182], DS175[183], DS175[186], DS175[187], DS175[189], DS175[190], DS175[192], DS175[194], DS175[196], DS175[197], DS175[198], DS175[201], DS175[202], DS175[203], DS175[204], DS175[205], DS175[206], DS175[207], DS175[208], DS175[209], DS175[210], DS175[212], DS175[213], DS175[214], DS175[215], DS175[216], DS175[218], DS175[219], DS175[220], DS175[221], DS175[222], DS175[223], DS175[225], DS175[226], DS175[227], DS175[228], DS175[229], DS175[230], DS175[231], DS175[232], DS175[233], DS175[234], DS175[235], DS175[237], DS175[238], DS175[239], DS175[240], DS175[241], DS175[243], DS175[244], DS175[245], DS175[246], DS175[247], DS175[248], DS175[249], DS175[250], DS175[251], DS175[252], DS175[254], DS175[255], DS175[256], DS175[257], DS175[258], DS175[259], DS175[262], DS175[265], DS175[267], DS175[268], DS175[270], DS175[271], DS175[272], DS175[273], DS175[274], DS175[275], DS175[276], DS175[279], DS175[280], DS175[282], DS175[283], DS175[285], DS175[288], DS175[290], DS175[297], DS175[298], DS175[302], DS175[307], DS175[308], DS175[309], DS175[312], DS175[313], DS175[314], DS175[316], DS175[318], DS175[319], DS175[320], DS175[321], DS175[322], DS175[324], DS175[325], DS175[326], DS175[327], DS175[334], DS175[335], DS175[336], DS175[338], DS175[339], DS175[340], DS175[341], DS175[342], DS175[343], DS175[346], DS175[349], DS175[350], DS175[353], DS175[354], DS175[355], DS175[356], DS175[357], DS175[358], DS175[359], DS175[360], DS175[361], DS175[362], DS175[364], DS175[365], DS175[366], DS175[367], DS175[368], DS175[370], DS175[371], DS175[372], DS175[373], DS175[374], DS175[377], DS175[378], DS175[379], DS175[380], DS175[381], DS175[382], DS175[383], DS175[384], DS175[385], DS175[386], DS175[387], DS175[388], DS175[389], DS175[390], DS175[391], DS175[392], DS175[393], DS175[394], DS175[395], DS175[396], DS175[400], DS175[401], DS175[402], DS175[403], DS175[404], DS175[405], DS175[407], DS175[408], DS175[409], DS175[410], DS175[411], DS175[412], DS175[413], DS175[415], DS175[416], DS175[417], DS175[418], DS175[419], DS175[420], DS175[422], DS175[423], DS175[424], DS175[425], DS175[426], DS175[427], DS175[428], DS175[429], DS175[430], DS175[431], DS175[432], DS175[434], DS175[436], DS175[437], DS175[438], DS175[439], DS175[440], DS175[441], DS175[442], DS175[443], DS175[444], DS175[445], DS175[446], DS175[447], DS175[448], DS175[449], DS175[450], DS175[451], DS175[452], DS175[453], DS175[454], DS175[455], DS175[456], DS175[457], DS175[459], DS175[460], DS175[461], DS175[463], DS175[464], DS175[465], DS175[466], DS175[467], DS175[468], DS175[469], DS175[470], DS175[472], DS175[473], DS175[475], DS175[476], DS175[477], DS175[479], DS175[482], DS175[484], DS175[485], DS175[486], DS175[487], DS175[488], DS175[489], DS175[490], DS175[491], DS175[492], DS175[493], DS175[494], DS175[495], DS175[496], DS175[497], DS175[498], DS175[499], DS175[500], DS175[501], DS175[502], DS175[503], DS175[505], DS175[506], DS175[507], DS175[508], DS175[509], DS175[510], DS175[513], DS175[514], DS175[517], DS175[518], DS175[520], DS175[521], DS175[522], DS175[523], DS175[524], DS175[527], DS175[528], DS175[529], DS175[531], DS175[532], DS175[533], DS175[534], DS175[535], DS175[536], DS175[537], DS175[538], DS175[539], DS175[540], DS175[541], DS175[542], DS175[543], DS175[544], DS175[545], DS175[546], DS175[547], DS175[548], DS175[549], DS175[550], DS175[551], DS175[552], DS175[553], DS175[554], DS175[555], DS175[556], DS175[557], DS175[558], DS175[559], DS175[560], DS175[561], DS175[562], DS175[563], DS175[564], DS175[565], DS175[566], DS175[567], DS175[568], DS175[569], DS175[570], DS175[571], DS175[572], DS175[573], DS175[574], DS175[575], DS175[576], DS175[577], DS175[578], DS175[579], DS175[580], DS175[581], DS175[582], DS175[583], DS175[584], DS175[585], DS175[586], DS175[587], DS175[588], DS175[589], DS175[590], DS175[591], DS175[592], DS175[593], DS175[594], DS175[595], DS175[596], DS175[597], DS175[598], DS175[599], DS175[600], DS175[601], DS175[602], DS175[603], DS175[604], DS175[605], DS175[606], DS175[607], DS175[608], DS175[609], DS175[610], DS175[611], DS175[612], DS175[613], DS175[614], DS175[615], DS175[616], DS175[617], DS175[618], DS175[619], DS175[620], DS175[621], DS175[622], DS175[623]. # Note that there is no reversible (64,28,12) difference sets in # group [64,175]. ########################### ########################### ###########################