### 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 [64,163]. Also, we determine the nonisomorphic # symmetric (64,28,12) designs that rise from these DSs. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,163]. # 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(64,163));' can be # used to create a list e of the elements of the group [64,163]. # In our list of (64,28,12) DSs in group [64,163], we have [1, # 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 18, 22, 23, # 29, 30, 34, 37, 40, 43, 49, 54, 57, 60, 62, 63, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[6], e[7], e[9], e[10], e[11], e[13], e[14], e[16], e[18], e[22], e[23], e[29], # e[30], e[34], e[37], e[40], e[43], e[49], e[54], e[57], e[60], e[62], e[63], e[64]] # where e is the list of the elements of the group [64,163] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,163));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 163] ### The number of inequivalent (64,28,12) DSs in group [64,163] is 920 ### The list of all inequivalent (64,28,12) DSs in group [64,163] is DS163:=[ [ 1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 18, 22, 23, 29, 30, 34, 37, 40, 43, 49, 54, 57, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 18, 22, 23, 29, 30, 34, 37, 40, 43, 44, 49, 54, 57, 58, 62, 63 ], [ 1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 18, 22, 29, 30, 34, 36, 40, 44, 49, 54, 57, 58, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 18, 22, 23, 29, 30, 34, 36, 40, 43, 45, 49, 54, 57, 59, 62, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 13, 15, 17, 19, 22, 23, 26, 32, 34, 37, 40, 43, 46, 52, 54, 56, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 13, 15, 17, 19, 22, 23, 26, 32, 34, 37, 40, 43, 44, 46, 52, 54, 56, 58, 61, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 22, 23, 26, 31, 37, 41, 43, 44, 46, 53, 54, 55, 56, 57, 58, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 22, 23, 26, 31, 34, 37, 41, 43, 44, 46, 54, 56, 57, 58, 61, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 26, 31, 37, 39, 43, 44, 46, 53, 54, 55, 56, 58, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 15, 17, 18, 22, 23, 26, 31, 34, 37, 39, 43, 44, 46, 54, 56, 58, 61, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 23, 26, 30, 34, 37, 40, 43, 46, 54, 56, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 23, 26, 30, 34, 37, 40, 43, 44, 46, 54, 56, 58, 61, 63 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 26, 31, 37, 39, 43, 46, 49, 50, 51, 52, 60, 61, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 26, 31, 37, 39, 43, 44, 46, 49, 50, 51, 52, 58, 61 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 26, 28, 31, 37, 39, 43, 46, 50, 52, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 26, 28, 31, 37, 39, 43, 44, 46, 50, 52, 58, 61, 62 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 25, 29, 31, 37, 39, 43, 47, 49, 52, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 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30, 37, 39, 43, 51, 53, 54, 55, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 28, 29, 30, 37, 39, 43, 44, 51, 53, 54, 55, 56, 57, 58 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 28, 29, 30, 34, 37, 39, 43, 51, 54, 56, 57, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 28, 29, 30, 34, 37, 39, 43, 44, 51, 54, 56, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 18, 19, 21, 23, 29, 30, 37, 41, 43, 49, 53, 54, 55, 56, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 18, 19, 21, 23, 29, 30, 37, 41, 43, 44, 49, 53, 54, 55, 56, 58, 62 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 18, 19, 21, 23, 29, 30, 34, 37, 41, 43, 49, 54, 56, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 18, 19, 21, 23, 29, 30, 34, 37, 41, 43, 44, 49, 54, 56, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 18, 21, 23, 29, 30, 34, 38, 40, 43, 49, 54, 57, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 18, 21, 23, 29, 30, 34, 38, 40, 43, 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5, 6, 7, 8, 9, 10, 12, 15, 17, 18, 19, 23, 26, 30, 36, 39, 47, 48, 51, 54, 56, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 18, 19, 23, 26, 30, 34, 36, 41, 44, 47, 48, 49, 54, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 18, 19, 20, 23, 26, 32, 35, 42, 46, 47, 53, 55, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 30, 32, 33, 37, 43, 45, 46, 49, 51, 52, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 27, 28, 30, 37, 45, 46, 48, 50, 51, 53, 54, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 19, 30, 32, 33, 34, 37, 40, 43, 46, 49, 52, 53, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 30, 32, 33, 37, 46, 52, 53, 55, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 18, 29, 30, 34, 35, 37, 43, 46, 48, 49, 50, 51, 56, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 23, 25, 38, 41, 43, 44, 46, 52, 54, 58, 59, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 23, 25, 27, 28, 38, 41, 46, 50, 51, 52, 54, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 25, 29, 33, 38, 43, 46, 49, 50, 51, 52, 60, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 27, 29, 30, 32, 34, 37, 39, 44, 45, 48, 51, 52, 56, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 25, 28, 30, 33, 37, 45, 46, 48, 53, 54, 55, 56, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 25, 27, 28, 30, 37, 45, 46, 48, 49, 50, 55, 56, 57, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 30, 32, 33, 34, 35, 37, 39, 45, 48, 51, 52, 56, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25, 30, 33, 34, 37, 44, 45, 46, 48, 53, 54, 55, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25, 27, 28, 30, 37, 44, 45, 46, 48, 50, 51, 53, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 19, 20, 26, 28, 32, 34, 40, 42, 43, 44, 46, 47, 53, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 18, 26, 30, 33, 38, 43, 44, 46, 47, 53, 54, 55, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 30, 32, 35, 38, 40, 43, 44, 46, 52, 54, 55, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 26, 30, 38, 39, 40, 43, 44, 46, 47, 55, 56, 57, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 26, 27, 31, 34, 35, 36, 43, 44, 47, 48, 49, 50, 57, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 23, 26, 28, 31, 36, 39, 43, 47, 48, 51, 54, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 23, 26, 27, 28, 31, 34, 36, 41, 44, 47, 48, 50, 54, 58, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 20, 22, 26, 31, 33, 35, 38, 41, 45, 46, 47, 49, 54, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 20, 22, 26, 31, 33, 35, 36, 39, 43, 47, 48, 51, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 20, 22, 26, 27, 31, 33, 38, 43, 44, 46, 47, 50, 51, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 20, 22, 23, 26, 31, 35, 36, 43, 45, 47, 48, 49, 51, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 20, 22, 23, 26, 31, 33, 38, 43, 45, 46, 47, 49, 51, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 20, 22, 26, 31, 34, 35, 36, 40, 43, 47, 48, 49, 53, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 17, 20, 22, 23, 26, 28, 31, 34, 36, 40, 41, 47, 48, 53, 54, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25, 27, 29, 30, 31, 32, 33, 37, 39, 44, 51, 53, 56, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 43, 44, 45, 46, 48, 50, 51, 52, 53, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 28, 43, 45, 46, 48, 49, 50, 52, 53, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 27, 44, 45, 46, 48, 49, 51, 52, 53, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 27, 28, 45, 46, 48, 52, 53, 58, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 30, 43, 44, 45, 46, 49, 50, 51, 53, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 28, 30, 43, 45, 46, 50, 53, 58, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 27, 45, 49, 51, 52, 53, 58, 59, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 24, 25, 27, 28, 44, 45, 52, 53, 59, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 25, 44, 50, 51, 52, 53, 58, 59, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 25, 30, 46, 49, 50, 51, 53, 58, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 25, 28, 49, 50, 52, 53, 59, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 19, 21, 22, 23, 24, 25, 28, 30, 44, 46, 50, 53, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17, 19, 25, 30, 31, 32, 33, 34, 37, 39, 44, 51, 53, 56, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 24, 26, 27, 31, 33, 41, 42, 44, 50, 51, 53, 59, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 22, 23, 24, 26, 31, 33, 41, 42, 45, 49, 51, 53, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 18, 19, 23, 24, 26, 31, 38, 39, 40, 41, 54, 56, 58, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 19, 22, 24, 26, 31, 33, 34, 41, 42, 45, 53, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 23, 27, 29, 30, 31, 32, 33, 37, 41, 49, 53, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 19, 23, 30, 31, 32, 33, 34, 37, 41, 49, 53, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 30, 31, 32, 34, 37, 45, 53, 54, 61, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 25, 26, 29, 34, 45, 49, 51, 52, 55, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 25, 26, 28, 29, 34, 44, 45, 52, 55, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 23, 25, 26, 34, 49, 50, 52, 55, 59, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 23, 25, 26, 28, 50, 51, 52, 53, 59, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 28, 29, 31, 33, 38, 41, 45, 46, 48, 49, 50, 53, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 27, 31, 32, 38, 41, 43, 44, 46, 50, 51, 52, 54, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 23, 31, 32, 38, 41, 43, 45, 46, 49, 51, 52, 54, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 31, 32, 34, 38, 40, 41, 45, 46, 49, 52, 53, 54, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 23, 31, 32, 38, 41, 46, 52, 53, 54, 55, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 18, 26, 27, 29, 33, 44, 46, 50, 51, 52, 53, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 26, 33, 34, 46, 50, 51, 52, 53, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 20, 22, 23, 26, 31, 35, 36, 39, 41, 47, 48, 54, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 20, 22, 23, 26, 31, 33, 38, 39, 41, 46, 47, 54, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 20, 22, 23, 26, 31, 34, 38, 39, 40, 46, 47, 55, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 21, 22, 24, 25, 30, 35, 39, 40, 44, 46, 50, 53, 55, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 21, 22, 24, 25, 30, 34, 35, 39, 40, 46, 50, 57, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 19, 21, 22, 24, 25, 30, 39, 41, 46, 50, 53, 54, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 25, 27, 29, 30, 31, 32, 37, 44, 45, 53, 56, 58, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 19, 21, 22, 24, 25, 30, 44, 46, 50, 53, 54, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 25, 30, 31, 32, 34, 37, 44, 45, 53, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 26, 33, 36, 38, 46, 50, 51, 52, 53, 54, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 26, 27, 29, 34, 36, 37, 46, 50, 51, 52, 58, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 26, 34, 36, 37, 46, 50, 51, 52, 53, 55, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 29, 30, 33, 38, 41, 44, 49, 50, 52, 53, 59, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 19, 30, 33, 38, 40, 41, 44, 49, 52, 53, 55, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 29, 30, 38, 43, 44, 50, 52, 53, 54, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 19, 30, 38, 40, 43, 44, 52, 53, 54, 55, 61, 62, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 508 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS163[2], DS163[5], DS163[8], DS163[9], DS163[12], DS163[14], DS163[17], DS163[18], DS163[20], DS163[23], DS163[24], DS163[25], DS163[29], DS163[31], DS163[36], DS163[43], DS163[47], DS163[51], DS163[53], DS163[54], DS163[55], DS163[56], DS163[57], DS163[58], DS163[60], DS163[61], DS163[64], DS163[67], DS163[68], DS163[70], DS163[74], DS163[75], DS163[77], DS163[80], DS163[81], DS163[83], DS163[84], DS163[85], DS163[86], DS163[87], DS163[88], DS163[89], DS163[90], DS163[91], DS163[95], DS163[97], DS163[98], DS163[105], DS163[108], DS163[109], DS163[110], DS163[111], DS163[112], DS163[113], DS163[114], DS163[120], DS163[121], DS163[122], DS163[125], DS163[128], DS163[131], DS163[132], DS163[136], DS163[137], DS163[138], DS163[139], DS163[142], DS163[145], DS163[146], DS163[147], DS163[160], DS163[161], DS163[166], DS163[170], DS163[173], DS163[185], DS163[187], DS163[188], DS163[189], DS163[193], DS163[194], DS163[197], DS163[198], DS163[200], DS163[204], DS163[205], DS163[218], DS163[223], DS163[226], DS163[227], DS163[230], DS163[242], DS163[244], DS163[247], DS163[250], DS163[251], DS163[269], DS163[271], DS163[272], DS163[275], DS163[277], DS163[283], DS163[284], DS163[286], DS163[287], DS163[288], DS163[291], DS163[294], DS163[296], DS163[298], DS163[302], DS163[306], DS163[308], DS163[312], DS163[313], DS163[314], DS163[318], DS163[319], DS163[320], DS163[321], DS163[322], DS163[324], DS163[327], DS163[334], DS163[340], DS163[341], DS163[342], DS163[343], DS163[344], DS163[345], DS163[346], DS163[347], DS163[348], DS163[349], DS163[350], DS163[352], DS163[353], DS163[354], DS163[356], DS163[357], DS163[358], DS163[359], DS163[360], DS163[361], DS163[362], DS163[363], DS163[364], DS163[365], DS163[367], DS163[368], DS163[369], DS163[372], DS163[374], DS163[376], DS163[378], DS163[383], DS163[384], DS163[388], DS163[391], DS163[392], DS163[397], DS163[398], DS163[400], DS163[401], DS163[402], DS163[403], DS163[404], DS163[406], DS163[408], DS163[410], DS163[413], DS163[414], DS163[415], DS163[416], DS163[417], DS163[418], DS163[420], DS163[424], DS163[425], DS163[426], DS163[428], DS163[429], DS163[430], DS163[432], DS163[435], DS163[436], DS163[437], DS163[440], DS163[443], DS163[445], DS163[446], DS163[447], DS163[448], DS163[450], DS163[455], DS163[456], DS163[460], DS163[461], DS163[463], DS163[464], DS163[466], DS163[468], DS163[469], DS163[471], DS163[472], DS163[473], DS163[474], DS163[477], DS163[478], DS163[479], DS163[481], DS163[482], DS163[483], DS163[485], DS163[486], DS163[487], DS163[497], DS163[502], DS163[507], DS163[508], DS163[509], DS163[511], DS163[512], DS163[513], DS163[515], DS163[516], DS163[518], DS163[519], DS163[520], DS163[523], DS163[524], DS163[525], DS163[528], DS163[538], DS163[539], DS163[543], DS163[547], DS163[549], DS163[553], DS163[555], DS163[556], DS163[557], DS163[558], DS163[560], DS163[561], DS163[562], DS163[565], DS163[567], DS163[568], DS163[569], DS163[571], DS163[574], DS163[575], DS163[576], DS163[580], DS163[581], DS163[582], DS163[587], DS163[588], DS163[590], DS163[592], DS163[593], DS163[594], DS163[596], DS163[597], DS163[598], DS163[599], DS163[600], DS163[601], DS163[602], DS163[604], DS163[605], DS163[606], DS163[607], DS163[608], DS163[610], DS163[616], DS163[617], DS163[619], DS163[620], DS163[627], DS163[642], DS163[644], DS163[648], DS163[649], DS163[659], DS163[661], DS163[662], DS163[663], DS163[664], DS163[665], DS163[666], DS163[669], DS163[670], DS163[673], DS163[674], DS163[675], DS163[676], DS163[677], DS163[678], DS163[679], DS163[680], DS163[681], DS163[682], DS163[683], DS163[684], DS163[685], DS163[686], DS163[687], DS163[688], DS163[689], DS163[690], DS163[691], DS163[692], DS163[693], DS163[697], DS163[698], DS163[699], DS163[700], DS163[701], DS163[702], DS163[703], DS163[704], DS163[705], DS163[706], DS163[707], DS163[708], DS163[709], DS163[710], DS163[711], DS163[712], DS163[713], DS163[715], DS163[716], DS163[717], DS163[718], DS163[719], DS163[720], DS163[721], DS163[722], DS163[723], DS163[724], DS163[725], DS163[726], DS163[727], DS163[728], DS163[729], DS163[730], DS163[731], DS163[732], DS163[733], DS163[734], DS163[735], DS163[736], DS163[737], DS163[738], DS163[739], DS163[740], DS163[742], DS163[743], DS163[744], DS163[745], DS163[749], DS163[750], DS163[753], DS163[755], DS163[756], DS163[757], DS163[758], DS163[759], DS163[762], DS163[763], DS163[764], DS163[765], DS163[766], DS163[768], DS163[769], DS163[770], DS163[773], DS163[774], DS163[775], DS163[776], DS163[777], DS163[784], DS163[785], DS163[786], DS163[787], DS163[788], DS163[789], DS163[790], DS163[791], DS163[792], DS163[793], DS163[794], DS163[795], DS163[796], DS163[797], DS163[798], DS163[799], DS163[800], DS163[801], DS163[802], DS163[803], DS163[804], DS163[805], DS163[806], DS163[807], DS163[808], DS163[809], DS163[810], DS163[811], DS163[812], DS163[813], DS163[814], DS163[815], DS163[816], DS163[817], DS163[818], DS163[819], DS163[820], DS163[821], DS163[822], DS163[823], DS163[824], DS163[825], DS163[826], DS163[827], DS163[828], DS163[829], DS163[830], DS163[831], DS163[832], DS163[833], DS163[834], DS163[835], DS163[836], DS163[837], DS163[838], DS163[839], DS163[840], DS163[841], DS163[842], DS163[843], DS163[844], DS163[845], DS163[846], DS163[847], DS163[848], DS163[849], DS163[850], DS163[851], DS163[852], DS163[853], DS163[854], DS163[855], DS163[856], DS163[857], DS163[858], DS163[859], DS163[860], DS163[861], DS163[862], DS163[863], DS163[864], DS163[865], DS163[866], DS163[867], DS163[868], DS163[869], DS163[870], DS163[871], DS163[872], DS163[873], DS163[874], DS163[876], DS163[886], DS163[887], DS163[888], DS163[889], DS163[890], DS163[891], DS163[892], DS163[893], DS163[894], DS163[895], DS163[896], DS163[897], DS163[898], DS163[899], DS163[900], DS163[901], DS163[902], DS163[903], DS163[904], DS163[905], DS163[906], DS163[907], DS163[908], DS163[909], DS163[911], DS163[913], DS163[914], DS163[915], DS163[916], DS163[917], DS163[918], DS163[919], DS163[920]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,163]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,163] is 36 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,163] is RDS163:=[ [ 2, 4, 5, 8, 9, 11, 17, 18, 20, 21, 23, 25, 29, 31, 34, 39, 42, 43, 47, 49, 52, 53, 55, 60, 61, 62, 63, 64 ], [ 2, 4, 5, 8, 9, 11, 15, 17, 18, 20, 21, 23, 25, 29, 31, 36, 38, 39, 42, 43, 47, 49, 52, 56, 60, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 11, 13, 17, 18, 20, 21, 23, 25, 29, 31, 33, 35, 39, 42, 43, 47, 49, 52, 54, 60, 61, 62, 64 ], [ 2, 3, 4, 8, 9, 11, 15, 16, 17, 18, 23, 25, 29, 31, 34, 38, 39, 43, 47, 49, 52, 53, 55, 60, 61, 62, 63, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 15, 16, 17, 18, 23, 25, 29, 31, 33, 35, 38, 39, 43, 47, 49, 52, 54, 60, 61, 62, 64 ], [ 2, 3, 4, 6, 8, 9, 11, 15, 16, 17, 18, 20, 22, 23, 25, 29, 31, 38, 39, 42, 43, 47, 49, 52, 60, 61, 62, 64 ], [ 2, 3, 4, 5, 9, 11, 16, 17, 18, 20, 21, 23, 31, 34, 36, 39, 42, 43, 49, 50, 51, 52, 53, 55, 56, 60, 63, 64 ], [ 2, 3, 4, 5, 9, 11, 13, 16, 17, 18, 20, 21, 23, 31, 33, 35, 36, 39, 42, 43, 49, 50, 51, 52, 54, 56, 60, 64 ], [ 2, 3, 4, 5, 8, 11, 16, 17, 18, 20, 21, 23, 25, 31, 34, 36, 39, 42, 43, 47, 52, 53, 55, 56, 60, 61, 63, 64 ], [ 2, 3, 4, 5, 8, 11, 13, 16, 17, 18, 20, 21, 23, 25, 31, 33, 35, 36, 39, 42, 43, 47, 52, 54, 56, 60, 61, 64 ], [ 2, 3, 4, 5, 8, 9, 11, 16, 17, 20, 21, 23, 24, 26, 31, 36, 41, 42, 43, 47, 49, 50, 51, 52, 56, 57, 60, 64 ], [ 2, 3, 4, 5, 8, 9, 11, 16, 17, 18, 20, 21, 23, 24, 26, 31, 36, 39, 42, 43, 47, 49, 50, 51, 52, 56, 60, 64 ], [ 2, 3, 4, 5, 8, 9, 11, 14, 16, 17, 18, 20, 21, 23, 25, 29, 31, 37, 39, 42, 43, 47, 49, 52, 60, 61, 62, 64 ], [ 2, 3, 4, 5, 6, 9, 11, 15, 16, 17, 18, 21, 22, 23, 31, 34, 38, 39, 43, 49, 50, 51, 52, 53, 55, 60, 63, 64 ], [ 2, 3, 4, 5, 6, 9, 11, 13, 15, 16, 17, 18, 21, 22, 23, 31, 33, 35, 38, 39, 43, 49, 50, 51, 52, 54, 60, 64 ], [ 2, 3, 4, 5, 6, 8, 11, 15, 16, 17, 18, 21, 22, 23, 25, 31, 34, 38, 39, 43, 47, 52, 53, 55, 60, 61, 63, 64 ], [ 2, 3, 4, 5, 6, 8, 11, 13, 15, 16, 17, 18, 21, 22, 23, 25, 31, 33, 35, 38, 39, 43, 47, 52, 54, 60, 61, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 15, 16, 17, 21, 22, 23, 24, 26, 31, 38, 41, 43, 47, 49, 50, 51, 52, 57, 60, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 15, 16, 17, 18, 21, 22, 23, 24, 26, 31, 38, 39, 43, 47, 49, 50, 51, 52, 60, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 17, 18, 22, 23, 25, 29, 31, 34, 39, 43, 47, 49, 52, 53, 55, 60, 61, 62, 63, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 15, 17, 18, 22, 23, 25, 29, 31, 36, 38, 39, 43, 47, 49, 52, 56, 60, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 13, 17, 18, 22, 23, 25, 29, 31, 33, 35, 39, 43, 47, 49, 52, 54, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 7, 9, 11, 15, 16, 17, 18, 20, 23, 31, 34, 38, 39, 42, 43, 49, 50, 51, 52, 53, 55, 60, 63, 64 ], [ 1, 2, 3, 4, 7, 9, 11, 13, 15, 16, 17, 18, 20, 23, 31, 33, 35, 38, 39, 42, 43, 49, 50, 51, 52, 54, 60, 64 ], [ 1, 2, 3, 4, 7, 8, 11, 15, 16, 17, 18, 20, 23, 25, 31, 34, 38, 39, 42, 43, 47, 52, 53, 55, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 7, 8, 11, 13, 15, 16, 17, 18, 20, 23, 25, 31, 33, 35, 38, 39, 42, 43, 47, 52, 54, 60, 61, 64 ], [ 1, 2, 3, 4, 7, 8, 9, 11, 15, 16, 17, 20, 23, 24, 26, 31, 38, 41, 42, 43, 47, 49, 50, 51, 52, 57, 60, 64 ], [ 1, 2, 3, 4, 7, 8, 9, 11, 15, 16, 17, 18, 20, 23, 24, 26, 31, 38, 39, 42, 43, 47, 49, 50, 51, 52, 60, 64 ], [ 1, 2, 3, 4, 6, 7, 9, 11, 16, 17, 18, 22, 23, 31, 34, 36, 39, 43, 49, 50, 51, 52, 53, 55, 56, 60, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 17, 18, 22, 23, 31, 33, 35, 36, 39, 43, 49, 50, 51, 52, 54, 56, 60, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 11, 16, 17, 18, 22, 23, 25, 31, 34, 36, 39, 43, 47, 52, 53, 55, 56, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 11, 13, 16, 17, 18, 22, 23, 25, 31, 33, 35, 36, 39, 43, 47, 52, 54, 56, 60, 61, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 16, 17, 22, 23, 24, 26, 31, 36, 41, 43, 47, 49, 50, 51, 52, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 16, 17, 18, 22, 23, 24, 26, 31, 36, 39, 43, 47, 49, 50, 51, 52, 56, 60, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 25, 29, 31, 37, 39, 43, 47, 49, 52, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 18, 21, 23, 25, 29, 31, 38, 39, 43, 47, 49, 52, 60, 61, 62, 64 ] ]; ########################### ########################### 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