### 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 [64,176]. 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 [64,176]. # 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(64,176));' can be # used to create a list e of the elements of the group [64,176]. # In our list of (64,28,12) DSs in group [64,176], we have [1, # 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 21, 23, 26, 30, # 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 55, 61, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[7], e[8], e[10], e[11], e[13], e[14], e[21], e[23], e[26], e[30], e[34], # e[35], e[36], e[38], e[39], e[40], e[41], e[43], e[44], e[46], e[55], e[61], e[64]] # where e is the list of the elements of the group [64,176] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,176));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 176] ### The number of inequivalent (64,28,12) DSs in group [64,176] is 379 ### The list of all inequivalent (64,28,12) DSs in group [64,176] is DS176:=[ [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 21, 23, 26, 30, 34, 35, 36, 38, 39, 40, 41, 43, 44, 46, 55, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 18, 21, 23, 26, 34, 35, 36, 38, 40, 46, 52, 55, 57, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 17, 21, 26, 28, 29, 30, 32, 36, 39, 41, 44, 45, 46, 51, 58, 59, 61 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 17, 21, 23, 26, 28, 29, 30, 32, 36, 39, 41, 43, 46, 51, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 17, 18, 21, 23, 26, 28, 29, 30, 32, 39, 43, 46, 51, 56, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 20, 25, 26, 28, 30, 31, 32, 33, 35, 36, 39, 40, 41, 48, 50, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 20, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 39, 40, 41, 48, 49, 54, 56 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 20, 25, 26, 28, 30, 31, 32, 33, 35, 36, 41, 48, 50, 54, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 20, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 41, 48, 49, 54, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 18, 20, 25, 26, 27, 30, 31, 32, 33, 35, 36, 39, 48, 51, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 19, 20, 25, 26, 28, 30, 31, 32, 33, 35, 38, 39, 48, 50, 54, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 19, 20, 25, 26, 27, 28, 30, 31, 32, 33, 35, 38, 39, 48, 49, 54, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 17, 19, 20, 25, 26, 28, 30, 31, 32, 35, 38, 40, 48, 50, 53, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 17, 19, 20, 25, 26, 27, 28, 30, 31, 32, 35, 38, 40, 48, 49, 53, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 17, 18, 20, 25, 26, 29, 30, 31, 32, 33, 35, 38, 39, 48, 49, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 17, 18, 20, 25, 26, 27, 29, 30, 31, 32, 35, 38, 39, 48, 50, 53, 56, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 20, 25, 26, 28, 30, 31, 32, 35, 37, 39, 40, 41, 48, 50, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 20, 25, 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37, 41, 42, 48, 50, 54, 55, 56, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18, 22, 23, 24, 29, 31, 33, 42, 44, 47, 48, 49, 50, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18, 22, 23, 24, 27, 29, 31, 42, 44, 47, 48, 53, 54, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18, 22, 23, 24, 26, 31, 33, 42, 43, 44, 49, 51, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18, 22, 23, 24, 26, 27, 31, 33, 42, 44, 50, 51, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 19, 25, 28, 31, 37, 38, 41, 43, 45, 48, 49, 50, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 19, 24, 25, 27, 32, 37, 38, 41, 45, 49, 50, 57, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 23, 24, 25, 29, 32, 37, 38, 39, 44, 50, 51, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 22, 27, 28, 30, 33, 34, 37, 41, 42, 48, 49, 54, 55, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 19, 27, 32, 36, 39, 43, 44, 47, 48, 49, 50, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 27, 28, 29, 32, 36, 39, 44, 47, 48, 57, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 22, 27, 28, 30, 33, 35, 36, 41, 42, 48, 49, 54, 56, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 22, 29, 30, 33, 35, 37, 41, 42, 48, 49, 54, 56, 58, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 22, 27, 29, 30, 35, 37, 41, 42, 48, 50, 53, 56, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 22, 26, 31, 37, 42, 43, 49, 51, 53, 54, 55, 56, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 18, 22, 26, 27, 31, 37, 42, 50, 51, 53, 54, 55, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 22, 27, 28, 30, 35, 37, 41, 42, 48, 49, 53, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 22, 29, 30, 33, 35, 36, 41, 42, 48, 49, 54, 58, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 22, 27, 29, 30, 35, 36, 41, 42, 48, 50, 53, 58, 59, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 184 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS176[1], DS176[6], DS176[10], DS176[11], DS176[12], DS176[13], DS176[14], DS176[22], DS176[23], DS176[24], DS176[31], DS176[33], DS176[37], DS176[40], DS176[43], DS176[44], DS176[45], DS176[46], DS176[48], DS176[53], DS176[54], DS176[57], DS176[61], DS176[64], DS176[65], DS176[67], DS176[68], DS176[71], DS176[72], DS176[76], DS176[77], DS176[78], DS176[81], DS176[82], DS176[83], DS176[85], DS176[88], DS176[89], DS176[90], DS176[93], DS176[97], DS176[98], DS176[101], DS176[102], DS176[105], DS176[106], DS176[107], DS176[110], DS176[111], DS176[112], DS176[116], DS176[117], DS176[118], DS176[119], DS176[120], DS176[124], DS176[125], DS176[126], DS176[129], DS176[133], DS176[137], DS176[138], DS176[140], DS176[141], DS176[145], DS176[146], DS176[148], DS176[149], DS176[152], DS176[153], DS176[156], DS176[157], DS176[160], DS176[161], DS176[162], DS176[164], DS176[165], DS176[167], DS176[168], DS176[169], DS176[171], DS176[172], DS176[173], DS176[175], DS176[177], DS176[178], DS176[181], DS176[184], DS176[185], DS176[186], DS176[188], DS176[190], DS176[191], DS176[193], DS176[194], DS176[195], DS176[200], DS176[201], DS176[203], DS176[206], DS176[207], DS176[208], DS176[215], DS176[217], DS176[218], DS176[219], DS176[223], DS176[227], DS176[228], DS176[229], DS176[235], DS176[236], DS176[239], DS176[240], DS176[241], DS176[242], DS176[243], DS176[244], DS176[245], DS176[246], DS176[247], DS176[248], DS176[249], DS176[251], DS176[252], DS176[253], DS176[256], DS176[260], DS176[261], DS176[264], DS176[266], DS176[267], DS176[268], DS176[269], DS176[276], DS176[278], DS176[279], DS176[280], DS176[282], DS176[284], DS176[285], DS176[286], DS176[295], DS176[297], DS176[298], DS176[304], DS176[309], DS176[310], DS176[311], DS176[314], DS176[318], DS176[319], DS176[320], DS176[324], DS176[326], DS176[327], DS176[328], DS176[332], DS176[333], DS176[334], DS176[335], DS176[339], DS176[341], DS176[349], DS176[351], DS176[352], DS176[353], DS176[355], DS176[356], DS176[358], DS176[359], DS176[362], DS176[363], DS176[364], DS176[365], DS176[366], DS176[367], DS176[368], DS176[370], DS176[371], DS176[372], DS176[373], DS176[375], DS176[376]. # Note that there is no reversible (64,28,12) difference sets in # group [64,176]. ########################### ########################### ###########################