### 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 [64,139]. 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 [64,139]. # 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(64,139));' can be # used to create a list e of the elements of the group [64,139]. # In our list of (64,28,12) DSs in group [64,139], we have [1, # 2, 3, 4, 5, 9, 11, 12, 14, 15, 20, 22, 25, 29, 30, # 32, 33, 35, 40, 41, 48, 52, 55, 56, 57, 58, 63, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[9], e[11], e[12], e[14], e[15], e[20], e[22], e[25], e[29], e[30], e[32], # e[33], e[35], e[40], e[41], e[48], e[52], e[55], e[56], e[57], e[58], e[63], e[64]] # where e is the list of the elements of the group [64,139] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,139));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 139] ### The number of inequivalent (64,28,12) DSs in group [64,139] is 751 ### The list of all inequivalent (64,28,12) DSs in group [64,139] is DS139:=[ [ 1, 2, 3, 4, 5, 9, 11, 12, 14, 15, 20, 22, 25, 29, 30, 32, 33, 35, 40, 41, 48, 52, 55, 56, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 12, 13, 15, 17, 18, 20, 22, 24, 25, 26, 27, 36, 37, 44, 47, 48, 50, 54, 55, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 32, 33, 35, 40, 41, 44, 46, 55, 56, 57, 58, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 30, 32, 33, 35, 40, 41, 44, 49, 52, 55, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 20, 22, 26, 28, 32, 33, 35, 40, 41, 46, 49, 51, 55, 56, 57, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 18, 20, 22, 26, 32, 33, 34, 35, 39, 40, 44, 46, 53, 56, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 14, 15, 18, 20, 22, 26, 28, 32, 33, 34, 35, 39, 40, 46, 49, 51, 53, 56, 61, 62 ], [ 1, 2, 3, 4, 5, 8, 11, 12, 13, 15, 17, 18, 20, 22, 24, 26, 32, 36, 37, 43, 44, 47, 54, 55, 57, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 17, 20, 22, 26, 29, 33, 34, 35, 37, 38, 39, 41, 43, 44, 53, 56, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 15, 17, 18, 20, 22, 26, 29, 33, 34, 35, 36, 37, 43, 44, 53, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 17, 18, 20, 22, 26, 29, 36, 37, 43, 44, 54, 55, 57, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 8, 9, 12, 13, 15, 17, 18, 20, 22, 25, 26, 27, 29, 30, 36, 37, 48, 50, 52, 54, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 16, 18, 19, 21, 31, 32, 36, 41, 43, 44, 45, 47, 48, 50, 51, 52, 56, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 13, 16, 18, 19, 22, 25, 27, 30, 36, 37, 43, 44, 54, 55, 57, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 19, 20, 21, 25, 30, 31, 34, 42, 43, 44, 45, 46, 47, 50, 51, 55, 58, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 19, 20, 21, 23, 24, 25, 27, 30, 31, 34, 42, 49, 51, 55, 59, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 18, 19, 21, 24, 25, 30, 31, 36, 41, 44, 45, 50, 51, 56, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 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17, 31, 32, 34, 37, 39, 40, 45, 49, 52, 55, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 17, 31, 32, 34, 37, 39, 40, 45, 46, 52, 55, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 17, 30, 31, 32, 34, 37, 39, 40, 45, 49, 55, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 17, 30, 31, 32, 34, 37, 39, 40, 45, 46, 55, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 17, 25, 29, 34, 37, 39, 40, 46, 47, 49, 52, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 25, 31, 33, 34, 35, 36, 37, 41, 45, 46, 55, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 21, 22, 29, 41, 44, 45, 46, 47, 50, 52, 60, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 19, 30, 31, 32, 34, 37, 40, 41, 45, 49, 55, 56, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 19, 29, 34, 37, 40, 41, 47, 48, 49, 52, 55, 56, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 31, 32, 33, 34, 35, 36, 37, 39, 41, 45, 52, 58, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 30, 31, 32, 33, 35, 36, 37, 41, 45, 46, 55, 57, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 21, 22, 35, 37, 41, 43, 44, 45, 46, 49, 52, 56, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 31, 32, 34, 36, 37, 41, 45, 49, 52, 55, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 31, 32, 34, 36, 37, 41, 45, 46, 52, 55, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 30, 31, 32, 34, 36, 37, 41, 45, 49, 55, 58, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 30, 31, 32, 34, 36, 37, 41, 45, 46, 55, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 19, 26, 27, 34, 36, 37, 41, 46, 49, 51, 52, 55, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 30, 34, 37, 39, 40, 44, 49, 52, 53, 54, 56, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 30, 33, 36, 37, 40, 43, 46, 52, 53, 55, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 30, 33, 35, 41, 43, 46, 52, 53, 54, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 30, 34, 35, 40, 43, 46, 52, 53, 55, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 30, 34, 41, 43, 46, 52, 53, 54, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 30, 33, 38, 40, 41, 44, 49, 52, 54, 56, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 30, 33, 35, 36, 38, 40, 43, 46, 52, 55, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 30, 34, 37, 39, 40, 44, 49, 52, 55, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19, 30, 34, 37, 40, 41, 44, 49, 52, 55, 56, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 30, 33, 35, 36, 37, 41, 43, 46, 52, 55, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 19, 30, 34, 36, 37, 41, 43, 46, 52, 55, 60, 62, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 405 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS139[1], DS139[14], DS139[21], DS139[25], DS139[28], DS139[42], DS139[59], DS139[72], DS139[79], DS139[80], DS139[82], DS139[87], DS139[101], DS139[103], DS139[104], DS139[109], DS139[114], DS139[118], DS139[122], DS139[124], DS139[127], DS139[131], DS139[134], DS139[135], DS139[137], DS139[141], DS139[145], DS139[146], DS139[149], DS139[154], DS139[158], DS139[163], DS139[164], DS139[165], DS139[166], DS139[167], DS139[169], DS139[170], DS139[172], DS139[174], DS139[182], DS139[183], DS139[203], DS139[205], DS139[206], DS139[207], DS139[215], DS139[216], DS139[217], DS139[218], DS139[221], DS139[223], DS139[235], DS139[238], DS139[244], DS139[245], DS139[249], DS139[251], DS139[256], DS139[257], DS139[258], DS139[266], DS139[267], DS139[268], DS139[269], DS139[270], DS139[271], DS139[273], DS139[274], DS139[275], DS139[276], DS139[277], DS139[278], DS139[284], DS139[285], DS139[286], DS139[287], DS139[289], DS139[290], DS139[299], DS139[300], DS139[301], DS139[302], DS139[303], DS139[304], DS139[306], DS139[307], DS139[308], DS139[310], DS139[311], DS139[312], DS139[313], DS139[314], DS139[315], DS139[317], DS139[319], DS139[322], DS139[323], DS139[324], DS139[326], DS139[329], DS139[336], DS139[338], DS139[340], DS139[341], DS139[343], DS139[344], DS139[345], DS139[346], DS139[356], DS139[357], DS139[358], DS139[359], DS139[360], DS139[361], DS139[362], DS139[363], DS139[364], DS139[365], DS139[367], DS139[368], DS139[370], DS139[372], DS139[373], DS139[375], DS139[377], DS139[378], DS139[379], DS139[380], DS139[381], DS139[383], DS139[384], DS139[386], DS139[387], DS139[388], DS139[390], DS139[392], DS139[395], DS139[396], DS139[397], DS139[398], DS139[400], DS139[402], DS139[403], DS139[404], DS139[405], DS139[406], DS139[407], DS139[408], DS139[414], DS139[417], DS139[418], DS139[419], DS139[421], DS139[423], DS139[424], DS139[425], DS139[427], DS139[428], DS139[429], DS139[430], DS139[431], DS139[432], DS139[433], DS139[434], DS139[435], DS139[436], DS139[437], DS139[438], DS139[439], DS139[440], DS139[441], DS139[442], DS139[443], DS139[444], DS139[445], DS139[446], DS139[447], DS139[448], DS139[449], DS139[451], DS139[453], DS139[454], DS139[455], DS139[468], DS139[470], DS139[471], DS139[473], DS139[476], DS139[477], DS139[479], DS139[480], DS139[481], DS139[482], DS139[483], DS139[484], DS139[485], DS139[486], DS139[490], DS139[492], DS139[493], DS139[494], DS139[495], DS139[497], DS139[501], DS139[502], DS139[503], DS139[504], DS139[505], DS139[506], DS139[507], DS139[508], DS139[512], DS139[513], DS139[514], DS139[515], DS139[516], DS139[520], DS139[525], DS139[527], DS139[528], DS139[530], DS139[532], DS139[533], DS139[535], DS139[539], DS139[540], DS139[541], DS139[542], DS139[545], DS139[546], DS139[548], DS139[549], DS139[550], DS139[551], DS139[552], DS139[553], DS139[554], DS139[555], DS139[556], DS139[557], DS139[559], DS139[560], DS139[561], DS139[562], DS139[563], DS139[564], DS139[565], DS139[566], DS139[568], DS139[569], DS139[570], DS139[572], DS139[574], DS139[575], DS139[576], DS139[577], DS139[578], DS139[579], DS139[580], DS139[582], DS139[583], DS139[584], DS139[585], DS139[586], DS139[587], DS139[588], DS139[589], DS139[590], DS139[591], DS139[592], DS139[593], DS139[594], DS139[595], DS139[596], DS139[597], DS139[598], DS139[599], DS139[600], DS139[601], DS139[602], DS139[603], DS139[604], DS139[605], DS139[606], DS139[607], DS139[608], DS139[609], DS139[611], DS139[612], DS139[613], DS139[614], DS139[615], DS139[616], DS139[617], DS139[618], DS139[619], DS139[620], DS139[621], DS139[622], DS139[623], DS139[624], DS139[625], DS139[626], DS139[627], DS139[628], DS139[629], DS139[630], DS139[631], DS139[632], DS139[633], DS139[634], DS139[635], DS139[636], DS139[637], DS139[638], DS139[639], DS139[640], DS139[641], DS139[642], DS139[643], DS139[644], DS139[645], DS139[646], DS139[647], DS139[648], DS139[650], DS139[651], DS139[658], DS139[659], DS139[660], DS139[661], DS139[662], DS139[663], DS139[664], DS139[665], DS139[666], DS139[667], DS139[668], DS139[669], DS139[670], DS139[671], DS139[672], DS139[673], DS139[674], DS139[675], DS139[676], DS139[677], DS139[678], DS139[679], DS139[680], DS139[681], DS139[682], DS139[683], DS139[684], DS139[685], DS139[686], DS139[687], DS139[688], DS139[689], DS139[690], DS139[691], DS139[693], DS139[694], DS139[696], DS139[697], DS139[698], DS139[700], DS139[701], DS139[704], DS139[705], DS139[707], DS139[708], DS139[709], DS139[710], DS139[712], DS139[713], DS139[714], DS139[716], DS139[717], DS139[718], DS139[719], DS139[721], DS139[722], DS139[724], DS139[726], DS139[727], DS139[728], DS139[729], DS139[730], DS139[731], DS139[733], DS139[734], DS139[737], DS139[738], DS139[739], DS139[741], DS139[742], DS139[745], DS139[746], DS139[747], DS139[748], DS139[749], DS139[750], DS139[751]. # Note that there is no reversible (64,28,12) difference sets in # group [64,139]. ########################### ########################### ###########################