### 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 [64,108]. 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 [64,108]. # 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(64,108));' can be # used to create a list e of the elements of the group [64,108]. # In our list of (64,28,12) DSs in group [64,108], we have [1, # 2, 3, 4, 5, 6, 8, 9, 16, 17, 18, 20, 25, 26, 30, # 31, 32, 34, 36, 39, 48, 49, 50, 51, 53, 55, 56, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[9], e[16], e[17], e[18], e[20], e[25], e[26], e[30], e[31], # e[32], e[34], e[36], e[39], e[48], e[49], e[50], e[51], e[53], e[55], e[56], e[63]] # where e is the list of the elements of the group [64,108] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,108));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 108] ### The number of inequivalent (64,28,12) DSs in group [64,108] is 602 ### The list of all inequivalent (64,28,12) DSs in group [64,108] is DS108:=[ [ 1, 2, 3, 4, 5, 6, 8, 9, 16, 17, 18, 20, 25, 26, 30, 31, 32, 34, 36, 39, 48, 49, 50, 51, 53, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 16, 17, 18, 20, 25, 26, 30, 31, 32, 33, 34, 37, 39, 48, 49, 50, 51, 54, 55 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 15, 16, 17, 18, 21, 23, 25, 30, 31, 32, 37, 38, 40, 44, 49, 50, 51, 53, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 18, 20, 21, 23, 25, 30, 31, 32, 33, 35, 36, 42, 44, 49, 50, 51, 54, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 18, 20, 21, 23, 25, 27, 29, 30, 31, 32, 33, 36, 42, 44, 50, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 19, 20, 25, 26, 28, 30, 31, 32, 33, 36, 40, 48, 50, 55, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 19, 20, 25, 26, 27, 28, 30, 31, 32, 33, 36, 40, 48, 49, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 18, 20, 25, 26, 30, 31, 32, 33, 35, 36, 39, 48, 49, 50, 51, 54, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 18, 20, 25, 26, 27, 29, 30, 31, 32, 33, 36, 39, 48, 50, 55, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 19, 20, 25, 26, 28, 30, 31, 32, 33, 37, 39, 48, 50, 55, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 19, 20, 25, 26, 27, 28, 30, 31, 32, 33, 37, 39, 48, 49, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 18, 21, 23, 25, 30, 31, 32, 35, 36, 37, 39, 44, 49, 50, 51, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 18, 20, 21, 23, 25, 30, 31, 32, 35, 37, 42, 44, 49, 50, 51, 53, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 18, 20, 21, 23, 25, 29, 30, 31, 32, 33, 37, 42, 44, 49, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 17, 19, 21, 25, 26, 30, 31, 32, 34, 36, 39, 48, 49, 50, 51, 53, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 17, 18, 20, 25, 26, 30, 31, 32, 35, 37, 39, 48, 49, 50, 51, 53, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 17, 18, 20, 25, 26, 29, 30, 31, 32, 33, 37, 39, 48, 49, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 17, 19, 20, 28, 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10, 11, 12, 13, 19, 24, 25, 27, 33, 36, 41, 46, 47, 48, 49, 57, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 23, 30, 31, 36, 39, 41, 44, 46, 47, 54, 58, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 26, 30, 33, 37, 41, 43, 46, 48, 52, 57, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 26, 30, 31, 33, 40, 41, 46, 48, 56, 58, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 28, 33, 37, 38, 42, 43, 46, 51, 54, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 27, 34, 37, 38, 42, 43, 46, 50, 55, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 21, 29, 33, 38, 39, 44, 51, 53, 54, 56, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 30, 33, 37, 38, 41, 43, 46, 52, 54, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 18, 27, 36, 37, 38, 41, 43, 46, 50, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 30, 34, 37, 38, 40, 43, 46, 52, 55, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 28, 36, 37, 38, 40, 43, 46, 51, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 31, 33, 36, 38, 41, 43, 47, 52, 53, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 27, 36, 37, 38, 41, 43, 47, 49, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 23, 30, 31, 36, 38, 39, 41, 47, 53, 54, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 31, 33, 34, 35, 39, 40, 49, 52, 56, 57, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 27, 30, 33, 34, 36, 37, 41, 46, 52, 55, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 25, 26, 30, 39, 41, 43, 44, 50, 52, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 27, 33, 36, 37, 38, 41, 46, 49, 57, 60, 62, 64 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 461 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS108[2], DS108[3], DS108[6], DS108[7], DS108[9], DS108[10], DS108[11], DS108[12], DS108[14], DS108[15], DS108[16], DS108[17], DS108[18], DS108[20], DS108[21], DS108[28], DS108[29], DS108[33], DS108[34], DS108[42], DS108[43], DS108[50], DS108[52], DS108[53], DS108[54], DS108[55], DS108[56], DS108[59], DS108[60], DS108[61], DS108[62], DS108[63], DS108[67], DS108[68], DS108[69], DS108[73], DS108[74], DS108[75], DS108[78], DS108[79], DS108[80], DS108[81], DS108[83], DS108[84], DS108[85], DS108[86], DS108[87], DS108[88], DS108[89], DS108[90], DS108[91], DS108[92], DS108[93], DS108[94], DS108[95], DS108[96], DS108[97], DS108[98], DS108[100], DS108[101], DS108[105], DS108[112], DS108[113], DS108[114], DS108[115], DS108[116], DS108[119], DS108[122], DS108[125], DS108[126], DS108[127], DS108[129], DS108[130], DS108[132], DS108[133], DS108[136], DS108[138], DS108[139], DS108[140], DS108[141], DS108[142], DS108[143], DS108[144], DS108[146], DS108[147], DS108[148], DS108[149], DS108[153], DS108[154], DS108[156], DS108[157], DS108[158], DS108[159], DS108[160], DS108[161], DS108[162], DS108[163], DS108[164], DS108[166], DS108[167], DS108[168], DS108[173], DS108[175], DS108[176], DS108[178], DS108[179], DS108[180], DS108[181], DS108[182], DS108[183], DS108[185], DS108[187], DS108[189], DS108[190], DS108[192], DS108[194], DS108[197], DS108[198], DS108[201], DS108[202], DS108[203], DS108[204], DS108[206], DS108[207], DS108[208], DS108[209], DS108[210], DS108[212], DS108[213], DS108[215], DS108[216], DS108[217], DS108[218], DS108[220], DS108[221], DS108[222], DS108[223], DS108[224], DS108[225], DS108[226], DS108[228], DS108[229], DS108[230], DS108[231], DS108[234], DS108[239], DS108[240], DS108[241], DS108[242], DS108[245], DS108[246], DS108[247], DS108[248], DS108[249], DS108[250], DS108[251], DS108[254], DS108[255], DS108[256], DS108[257], DS108[258], DS108[259], DS108[260], DS108[261], DS108[262], DS108[263], DS108[265], DS108[266], DS108[267], DS108[268], DS108[269], DS108[270], DS108[271], DS108[272], DS108[273], DS108[275], DS108[279], DS108[280], DS108[281], DS108[282], DS108[283], DS108[284], DS108[286], DS108[287], DS108[288], DS108[289], DS108[290], DS108[292], DS108[293], DS108[294], DS108[298], DS108[300], DS108[303], DS108[305], DS108[306], DS108[308], DS108[310], DS108[311], DS108[320], DS108[321], DS108[322], DS108[323], DS108[324], DS108[325], DS108[326], DS108[327], DS108[330], DS108[331], DS108[332], DS108[333], DS108[334], DS108[335], DS108[336], DS108[337], DS108[339], DS108[340], DS108[342], DS108[343], DS108[344], DS108[345], DS108[346], DS108[347], DS108[348], DS108[349], DS108[350], DS108[351], DS108[352], DS108[353], DS108[354], DS108[355], DS108[356], DS108[357], DS108[358], DS108[359], DS108[360], DS108[361], DS108[362], DS108[363], DS108[364], DS108[365], DS108[366], DS108[367], DS108[368], DS108[369], DS108[370], DS108[371], DS108[373], DS108[374], DS108[375], DS108[376], DS108[377], DS108[378], DS108[379], DS108[380], DS108[381], DS108[382], DS108[383], DS108[384], DS108[385], DS108[386], DS108[387], DS108[389], DS108[390], DS108[391], DS108[392], DS108[393], DS108[394], DS108[395], DS108[396], DS108[397], DS108[398], DS108[400], DS108[401], DS108[402], DS108[403], DS108[404], DS108[405], DS108[406], DS108[407], DS108[408], DS108[409], DS108[410], DS108[411], DS108[412], DS108[413], DS108[414], DS108[415], DS108[416], DS108[417], DS108[418], DS108[419], DS108[420], DS108[421], DS108[422], DS108[423], DS108[424], DS108[425], DS108[426], DS108[427], DS108[428], DS108[429], DS108[430], DS108[432], DS108[433], DS108[434], DS108[435], DS108[436], DS108[437], DS108[439], DS108[442], DS108[443], DS108[444], DS108[445], DS108[446], DS108[447], DS108[448], DS108[449], DS108[450], DS108[451], DS108[453], DS108[454], DS108[455], DS108[456], DS108[457], DS108[458], DS108[459], DS108[460], DS108[461], DS108[462], DS108[463], DS108[464], DS108[465], DS108[466], DS108[467], DS108[468], DS108[469], DS108[470], DS108[472], DS108[475], DS108[476], DS108[477], DS108[478], DS108[479], DS108[480], DS108[481], DS108[482], DS108[483], DS108[484], DS108[485], DS108[486], DS108[487], DS108[488], DS108[489], DS108[490], DS108[491], DS108[492], DS108[493], DS108[494], DS108[495], DS108[496], DS108[497], DS108[498], DS108[499], DS108[500], DS108[501], DS108[502], DS108[503], DS108[504], DS108[505], DS108[506], DS108[507], DS108[508], DS108[509], DS108[510], DS108[511], DS108[512], DS108[513], DS108[514], DS108[515], DS108[516], DS108[517], DS108[518], DS108[519], DS108[520], DS108[522], DS108[523], DS108[524], DS108[526], DS108[527], DS108[528], DS108[529], DS108[530], DS108[531], DS108[532], DS108[533], DS108[534], DS108[535], DS108[536], DS108[537], DS108[538], DS108[539], DS108[540], DS108[541], DS108[543], DS108[544], DS108[545], DS108[546], DS108[547], DS108[548], DS108[551], DS108[552], DS108[553], DS108[554], DS108[555], DS108[556], DS108[557], DS108[558], DS108[559], DS108[560], DS108[561], DS108[562], DS108[563], DS108[564], DS108[565], DS108[566], DS108[567], DS108[568], DS108[569], DS108[570], DS108[571], DS108[572], DS108[573], DS108[574], DS108[575], DS108[576], DS108[577], DS108[578], DS108[579], DS108[580], DS108[581], DS108[582], DS108[583], DS108[584], DS108[585], DS108[586], DS108[587], DS108[588], DS108[589], DS108[590], DS108[591], DS108[592], DS108[593], DS108[594], DS108[595], DS108[596], DS108[597], DS108[598], DS108[599], DS108[600], DS108[601], DS108[602]. # Note that there is no reversible (64,28,12) difference sets in # group [64,108]. ########################### ########################### ###########################