### 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 [64,157]. 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 [64,157]. # 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(64,157));' can be # used to create a list e of the elements of the group [64,157]. # In our list of (64,28,12) DSs in group [64,157], we have [1, # 2, 3, 4, 5, 6, 10, 11, 13, 18, 20, 23, 28, 30, 35, # 36, 37, 38, 40, 44, 45, 49, 51, 53, 57, 60, 62, 63]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[10], e[11], e[13], e[18], e[20], e[23], e[28], e[30], e[35], e[36], # e[37], e[38], e[40], e[44], e[45], e[49], e[51], e[53], e[57], e[60], e[62], e[63]] # where e is the list of the elements of the group [64,157] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,157));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 157] ### The number of inequivalent (64,28,12) DSs in group [64,157] is 670 ### The list of all inequivalent (64,28,12) DSs in group [64,157] is DS157:=[ [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 18, 20, 23, 28, 30, 35, 36, 37, 38, 40, 44, 45, 49, 51, 53, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 18, 20, 23, 28, 30, 34, 35, 36, 37, 38, 40, 45, 49, 51, 55, 57, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 16, 18, 20, 23, 28, 30, 34, 36, 40, 44, 45, 49, 51, 54, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 15, 18, 20, 23, 28, 30, 35, 37, 40, 44, 45, 49, 51, 53, 56, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 15, 18, 20, 23, 28, 30, 34, 35, 37, 40, 45, 49, 51, 55, 56, 57, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 10, 11, 13, 15, 16, 18, 20, 23, 28, 30, 34, 38, 40, 45, 49, 51, 54, 57, 58, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 23, 28, 29, 30, 34, 38, 40, 43, 45, 51, 52, 54, 56, 57, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 18, 20, 23, 27, 29, 30, 34, 38, 40, 44, 45, 50, 52, 54, 56, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 17, 19, 20, 23, 28, 30, 34, 35, 38, 40, 44, 50, 52, 55, 56, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 14, 19, 20, 23, 28, 29, 31, 34, 38, 39, 44, 51, 52, 53, 55, 56, 57, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 14, 18, 19, 20, 23, 29, 31, 34, 38, 41, 44, 49, 52, 53, 55, 56, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 19, 20, 23, 27, 29, 31, 34, 35, 38, 39, 44, 50, 52, 55, 56, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 18, 19, 20, 23, 27, 29, 31, 35, 38, 41, 44, 50, 52, 53, 56, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 29, 31, 34, 35, 38, 40, 44, 49, 52, 55, 56, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 17, 19, 20, 23, 28, 29, 31, 35, 38, 40, 44, 51, 52, 53, 56, 59, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 18, 20, 23, 29, 32, 34, 38, 40, 43, 45, 49, 54, 56, 57, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 18, 20, 23, 27, 29, 32, 34, 38, 40, 45, 50, 54, 56, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 9, 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10, 11, 14, 16, 23, 25, 28, 29, 31, 36, 46, 48, 49, 50, 51, 53, 54, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 23, 25, 28, 29, 31, 34, 35, 36, 46, 48, 50, 51, 57, 58, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 19, 30, 31, 33, 34, 36, 40, 46, 52, 53, 57, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 31, 32, 36, 40, 44, 48, 51, 52, 53, 54, 55, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 28, 31, 32, 34, 35, 36, 40, 48, 52, 55, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 25, 31, 36, 40, 41, 44, 46, 48, 51, 53, 54, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 25, 28, 31, 34, 35, 36, 40, 41, 46, 48, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19, 25, 31, 35, 36, 39, 45, 46, 48, 51, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 18, 19, 25, 28, 31, 36, 39, 45, 46, 48, 54, 57, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 25, 29, 30, 31, 32, 37, 39, 43, 44, 50, 51, 54, 58, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 25, 28, 30, 31, 32, 37, 39, 43, 45, 49, 51, 54, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 25, 28, 30, 31, 32, 35, 37, 39, 43, 44, 45, 51, 55, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 25, 28, 29, 30, 31, 32, 35, 37, 39, 43, 44, 50, 55, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 19, 25, 28, 30, 31, 32, 33, 34, 37, 40, 53, 57, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 25, 28, 30, 31, 32, 33, 35, 37, 39, 54, 57, 59, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 22, 24, 25, 26, 28, 31, 35, 39, 42, 45, 54, 55, 56, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 18, 20, 22, 24, 28, 31, 35, 36, 39, 45, 46, 47, 54, 55, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 19, 21, 22, 24, 30, 39, 41, 48, 50, 53, 54, 58, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 18, 19, 21, 22, 24, 30, 44, 48, 50, 53, 54, 57, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 24, 25, 26, 28, 31, 34, 35, 36, 45, 53, 54, 55, 56, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 19, 25, 30, 31, 32, 33, 36, 39, 40, 45, 51, 54, 56, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 19, 25, 28, 30, 31, 32, 33, 35, 36, 39, 40, 45, 55, 56, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 18, 25, 30, 31, 32, 35, 36, 40, 44, 51, 54, 55, 56, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 26, 29, 32, 34, 40, 48, 54, 57, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 25, 29, 30, 40, 41, 47, 53, 54, 58, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 25, 29, 30, 40, 41, 44, 47, 53, 54, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 25, 29, 30, 34, 40, 41, 47, 54, 55, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 25, 29, 30, 34, 40, 41, 44, 47, 54, 55, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 20, 21, 25, 26, 29, 32, 34, 40, 46, 54, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 29, 30, 35, 46, 47, 48, 53, 55, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 29, 30, 35, 44, 46, 47, 48, 53, 55, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 29, 30, 34, 35, 46, 47, 48, 57, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 29, 30, 34, 35, 44, 46, 47, 48, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 19, 30, 31, 34, 36, 40, 46, 49, 52, 53, 55, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 25, 30, 31, 32, 34, 37, 40, 44, 51, 53, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 25, 30, 31, 32, 35, 37, 39, 44, 51, 54, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 30, 32, 35, 36, 39, 44, 46, 49, 51, 55, 56, 58, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 30, 32, 34, 36, 39, 41, 46, 53, 54, 56, 57, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 30, 32, 34, 35, 36, 39, 41, 46, 49, 53, 55, 56, 57, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 28, 30, 32, 36, 39, 46, 49, 54, 56, 58, 59, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 19, 30, 32, 33, 35, 36, 39, 44, 45, 46, 54, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 27, 29, 32, 36, 39, 44, 45, 49, 52, 54, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 27, 29, 32, 35, 36, 39, 44, 45, 52, 55, 58, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 29, 36, 46, 49, 51, 53, 54, 57, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 29, 36, 39, 44, 45, 46, 49, 51, 54, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 29, 30, 36, 44, 51, 52, 53, 54, 57, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 28, 29, 35, 36, 39, 45, 46, 49, 55, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 28, 29, 34, 35, 36, 46, 49, 57, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 25, 27, 28, 29, 30, 34, 35, 36, 44, 52, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 32, 36, 39, 45, 49, 52, 53, 54, 55, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 32, 34, 36, 49, 52, 53, 54, 57, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 32, 34, 35, 36, 52, 53, 55, 57, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 32, 34, 35, 36, 39, 44, 45, 52, 55, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 25, 36, 39, 41, 45, 46, 49, 53, 54, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 25, 34, 35, 36, 39, 41, 44, 45, 46, 57, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 25, 30, 34, 36, 44, 51, 52, 53, 54, 57, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 25, 28, 30, 34, 35, 36, 52, 53, 55, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 25, 36, 39, 46, 49, 54, 57, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 25, 35, 36, 39, 46, 55, 57, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 27, 30, 32, 35, 37, 39, 44, 46, 50, 51, 55, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 27, 28, 30, 32, 37, 39, 46, 50, 54, 58, 59, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 30, 32, 34, 37, 40, 41, 46, 53, 54, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 30, 32, 34, 35, 37, 40, 41, 46, 49, 53, 55, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 25, 33, 36, 40, 46, 49, 53, 54, 57, 59, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 25, 33, 36, 39, 40, 45, 46, 49, 54, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 19, 20, 22, 25, 35, 36, 39, 46, 54, 55, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 19, 20, 22, 25, 34, 35, 36, 44, 45, 46, 54, 57, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 30, 32, 33, 36, 39, 40, 46, 54, 56, 58, 59, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 19, 30, 32, 33, 35, 36, 39, 40, 46, 49, 55, 56, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 30, 32, 35, 36, 39, 45, 46, 49, 54, 55, 56, 60 ] ]; ### 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: ### DS157[2], DS157[3], DS157[6], DS157[7], DS157[10], DS157[11], DS157[12], DS157[13], DS157[14], DS157[15], DS157[16], DS157[17], DS157[18], DS157[19], DS157[20], DS157[24], DS157[25], DS157[26], DS157[27], DS157[28], DS157[29], DS157[30], DS157[32], DS157[33], DS157[34], DS157[35], DS157[36], DS157[38], DS157[40], DS157[41], DS157[42], DS157[44], DS157[47], DS157[49], DS157[52], DS157[54], DS157[55], DS157[57], DS157[58], DS157[59], DS157[60], DS157[61], DS157[63], DS157[66], DS157[67], DS157[68], DS157[69], DS157[70], DS157[71], DS157[72], DS157[73], DS157[74], DS157[76], DS157[79], DS157[80], DS157[81], DS157[82], DS157[83], DS157[84], DS157[87], DS157[90], DS157[92], DS157[94], DS157[98], DS157[100], DS157[101], DS157[103], DS157[104], DS157[105], DS157[109], DS157[111], DS157[113], DS157[122], DS157[123], DS157[124], DS157[125], DS157[127], DS157[128], DS157[132], DS157[133], DS157[134], DS157[135], DS157[136], DS157[137], DS157[140], DS157[142], DS157[143], DS157[144], DS157[145], DS157[146], DS157[149], DS157[150], DS157[151], DS157[152], DS157[153], DS157[154], DS157[156], DS157[159], DS157[164], DS157[165], DS157[166], DS157[169], DS157[170], DS157[172], DS157[175], DS157[176], DS157[177], DS157[178], DS157[179], DS157[180], DS157[181], DS157[182], DS157[183], DS157[184], DS157[186], DS157[188], DS157[189], DS157[190], DS157[191], DS157[192], DS157[193], DS157[195], DS157[196], DS157[197], DS157[198], DS157[199], DS157[200], DS157[202], DS157[203], DS157[204], DS157[207], DS157[210], DS157[211], DS157[213], DS157[214], DS157[216], DS157[217], DS157[218], DS157[219], DS157[221], DS157[222], DS157[224], DS157[226], DS157[229], DS157[230], DS157[231], DS157[233], DS157[234], DS157[236], DS157[239], DS157[240], DS157[241], DS157[242], DS157[243], DS157[244], DS157[245], DS157[246], DS157[247], DS157[248], DS157[249], DS157[250], DS157[251], DS157[252], DS157[253], DS157[254], DS157[255], DS157[256], DS157[259], DS157[260], DS157[261], DS157[262], DS157[263], DS157[264], DS157[265], DS157[266], DS157[267], DS157[268], DS157[269], DS157[270], DS157[271], DS157[272], DS157[274], DS157[275], DS157[276], DS157[277], DS157[278], DS157[279], DS157[280], DS157[281], DS157[282], DS157[283], DS157[284], DS157[287], DS157[289], DS157[290], DS157[291], DS157[292], DS157[293], DS157[295], DS157[298], DS157[300], DS157[301], DS157[303], DS157[304], DS157[306], DS157[307], DS157[309], DS157[310], DS157[311], DS157[313], DS157[314], DS157[315], DS157[316], DS157[318], DS157[319], DS157[320], DS157[321], DS157[323], DS157[326], DS157[327], DS157[330], DS157[331], DS157[333], DS157[334], DS157[335], DS157[336], DS157[337], DS157[338], DS157[339], DS157[340], DS157[341], DS157[345], DS157[346], DS157[349], DS157[351], DS157[352], DS157[355], DS157[356], DS157[358], DS157[359], DS157[360], DS157[361], DS157[362], DS157[363], DS157[364], DS157[365], DS157[369], DS157[370], DS157[371], DS157[372], DS157[373], DS157[374], DS157[376], DS157[377], DS157[379], DS157[380], DS157[381], DS157[382], DS157[383], DS157[384], DS157[385], DS157[387], DS157[388], DS157[390], DS157[391], DS157[392], DS157[393], DS157[394], DS157[396], DS157[397], DS157[398], DS157[399], DS157[400], DS157[401], DS157[402], DS157[403], DS157[404], DS157[405], DS157[406], DS157[408], DS157[409], DS157[410], DS157[411], DS157[412], DS157[413], DS157[415], DS157[417], DS157[418], DS157[419], DS157[420], DS157[421], DS157[422], DS157[425], DS157[427], DS157[428], DS157[438], DS157[440], DS157[441], DS157[443], DS157[444], DS157[445], DS157[447], DS157[448], DS157[449], DS157[450], DS157[453], DS157[457], DS157[458], DS157[460], DS157[462], DS157[464], DS157[466], DS157[467], DS157[469], DS157[470], DS157[473], DS157[479], DS157[480], DS157[487], DS157[488], DS157[490], DS157[491], DS157[493], DS157[495], DS157[497], DS157[501], DS157[503], DS157[506], DS157[511], DS157[514], DS157[517], DS157[519], DS157[520], DS157[525], DS157[526], DS157[527], DS157[530], DS157[531], DS157[532], DS157[533], DS157[534], DS157[535], DS157[536], DS157[538], DS157[539], DS157[540], DS157[541], DS157[542], DS157[543], DS157[544], DS157[545], DS157[546], DS157[547], DS157[548], DS157[549], DS157[550], DS157[551], DS157[552], DS157[553], DS157[554], DS157[555], DS157[556], DS157[557], DS157[558], DS157[559], DS157[560], DS157[561], DS157[562], DS157[563], DS157[564], DS157[565], DS157[566], DS157[567], DS157[568], DS157[569], DS157[570], DS157[571], DS157[572], DS157[573], DS157[575], DS157[576], DS157[577], DS157[578], DS157[579], DS157[580], DS157[581], DS157[582], DS157[583], DS157[584], DS157[585], DS157[586], DS157[587], DS157[588], DS157[589], DS157[590], DS157[591], DS157[592], DS157[593], DS157[594], DS157[595], DS157[596], DS157[597], DS157[598], DS157[599], DS157[600], DS157[601], DS157[602], DS157[603], DS157[604], DS157[605], DS157[606], DS157[607], DS157[608], DS157[609], DS157[610], DS157[611], DS157[612], DS157[613], DS157[614], DS157[615], DS157[616], DS157[617], DS157[618], DS157[619], DS157[620], DS157[621], DS157[622], DS157[623], DS157[624], DS157[625], DS157[626], DS157[627], DS157[628], DS157[629], DS157[630], DS157[631], DS157[632], DS157[633], DS157[634], DS157[635], DS157[636], DS157[637], DS157[638], DS157[639], DS157[640], DS157[641], DS157[642], DS157[643], DS157[644], DS157[645], DS157[646], DS157[647], DS157[648], DS157[649], DS157[650], DS157[651], DS157[652], DS157[653], DS157[654], DS157[655], DS157[656], DS157[657], DS157[658], DS157[659], DS157[660], DS157[661], DS157[662], DS157[663], DS157[664], DS157[665], DS157[666], DS157[667], DS157[668], DS157[669], DS157[670]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,157]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,157] is 84 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,157] is RDS157:=[ [ 2, 5, 8, 9, 11, 13, 15, 20, 21, 23, 25, 27, 31, 33, 34, 36, 38, 42, 45, 47, 51, 52, 53, 56, 58, 61, 62, 64 ], [ 2, 4, 8, 9, 11, 13, 15, 18, 19, 23, 25, 27, 31, 33, 36, 38, 41, 45, 47, 51, 52, 55, 56, 58, 61, 62, 63, 64 ], [ 2, 4, 8, 9, 11, 13, 15, 18, 19, 23, 25, 27, 31, 33, 34, 36, 38, 41, 45, 47, 51, 52, 53, 56, 58, 61, 62, 64 ], [ 2, 4, 6, 8, 9, 11, 13, 18, 19, 20, 22, 23, 25, 27, 31, 33, 41, 42, 45, 47, 51, 52, 55, 58, 61, 62, 63, 64 ], [ 2, 4, 6, 8, 9, 11, 13, 18, 19, 20, 22, 23, 25, 27, 31, 33, 34, 41, 42, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 2, 4, 5, 9, 11, 13, 15, 19, 20, 21, 23, 27, 31, 33, 34, 36, 38, 39, 42, 44, 51, 52, 53, 56, 57, 59, 62, 64 ], [ 2, 4, 5, 9, 11, 13, 15, 17, 18, 20, 21, 23, 29, 31, 33, 36, 38, 39, 42, 44, 49, 52, 55, 56, 59, 62, 63, 64 ], [ 2, 4, 5, 8, 9, 13, 17, 19, 20, 21, 23, 27, 30, 31, 32, 33, 34, 40, 42, 45, 46, 47, 48, 51, 53, 58, 62, 64 ], [ 2, 4, 5, 8, 9, 11, 17, 18, 20, 21, 23, 25, 27, 31, 34, 39, 42, 45, 47, 51, 52, 53, 55, 58, 61, 62, 63, 64 ], [ 2, 4, 5, 8, 9, 11, 15, 17, 18, 20, 21, 23, 25, 27, 31, 36, 38, 39, 42, 45, 47, 51, 52, 56, 58, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 11, 13, 17, 19, 20, 21, 23, 25, 27, 31, 33, 34, 40, 42, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 11, 13, 17, 18, 20, 21, 23, 25, 27, 31, 33, 35, 39, 42, 45, 47, 51, 52, 54, 58, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 11, 13, 15, 19, 20, 21, 25, 27, 31, 33, 34, 36, 38, 39, 42, 47, 51, 52, 53, 56, 57, 61, 62 ], [ 2, 4, 5, 8, 9, 11, 13, 15, 17, 18, 20, 21, 25, 29, 31, 33, 36, 38, 39, 42, 47, 49, 52, 55, 56, 61, 62, 63 ], [ 2, 4, 5, 8, 9, 10, 12, 13, 19, 20, 21, 23, 25, 28, 31, 33, 39, 42, 43, 47, 50, 55, 57, 60, 61, 62, 63, 64 ], [ 2, 4, 5, 8, 9, 10, 12, 13, 19, 20, 21, 23, 25, 28, 31, 33, 34, 39, 42, 43, 47, 50, 53, 57, 60, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 10, 12, 13, 17, 20, 21, 23, 25, 28, 31, 33, 34, 41, 42, 45, 47, 50, 53, 57, 58, 61, 62, 64 ], [ 2, 4, 5, 8, 9, 10, 12, 13, 17, 18, 20, 21, 23, 25, 28, 31, 33, 39, 42, 45, 47, 50, 55, 58, 61, 62, 63, 64 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 29, 31, 33, 36, 38, 41, 44, 45, 47, 49, 52, 55, 58, 59, 62, 63 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 27, 31, 33, 34, 36, 38, 41, 43, 44, 47, 51, 52, 53, 59, 60, 62 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 23, 29, 31, 33, 36, 38, 41, 43, 47, 49, 52, 55, 60, 62, 63, 64 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 23, 29, 31, 33, 34, 36, 38, 41, 43, 47, 49, 52, 53, 60, 62, 64 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 23, 27, 31, 33, 36, 38, 41, 45, 47, 51, 52, 55, 58, 62, 63, 64 ], [ 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 18, 19, 23, 27, 31, 33, 34, 36, 38, 41, 45, 47, 51, 52, 53, 58, 62, 64 ], [ 2, 4, 5, 6, 9, 11, 13, 15, 18, 19, 21, 22, 23, 27, 31, 33, 34, 36, 38, 41, 44, 51, 52, 53, 56, 59, 62, 64 ], [ 2, 4, 5, 6, 8, 9, 11, 13, 15, 18, 19, 21, 22, 25, 27, 31, 33, 34, 36, 38, 41, 47, 51, 52, 53, 56, 61, 62 ], [ 2, 4, 5, 6, 8, 9, 10, 12, 13, 18, 19, 21, 22, 23, 25, 28, 31, 33, 41, 43, 47, 50, 55, 60, 61, 62, 63, 64 ], [ 2, 4, 5, 6, 8, 9, 10, 12, 13, 18, 19, 21, 22, 23, 25, 28, 31, 33, 34, 41, 43, 47, 50, 53, 60, 61, 62, 64 ], [ 2, 4, 5, 6, 7, 9, 13, 15, 18, 19, 20, 22, 23, 25, 29, 31, 33, 34, 41, 42, 43, 49, 53, 56, 60, 61, 62, 64 ], [ 2, 4, 5, 6, 7, 9, 13, 15, 18, 19, 20, 22, 23, 25, 27, 31, 33, 41, 42, 45, 51, 55, 56, 58, 61, 62, 63, 64 ], [ 2, 4, 5, 6, 7, 9, 13, 15, 18, 19, 20, 22, 23, 25, 27, 31, 33, 34, 41, 42, 45, 51, 53, 56, 58, 61, 62, 64 ], [ 2, 3, 5, 8, 9, 11, 13, 14, 16, 20, 21, 23, 25, 27, 31, 33, 34, 37, 42, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 14, 16, 18, 19, 23, 25, 27, 31, 33, 37, 41, 45, 47, 51, 52, 55, 58, 61, 62, 63, 64 ], [ 2, 3, 4, 8, 9, 11, 13, 14, 16, 18, 19, 23, 25, 27, 31, 33, 34, 37, 41, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 2, 3, 4, 5, 9, 11, 13, 14, 16, 19, 20, 21, 23, 27, 31, 33, 34, 37, 39, 42, 44, 51, 52, 53, 57, 59, 62, 64 ], [ 2, 3, 4, 5, 9, 11, 13, 14, 16, 17, 18, 20, 21, 23, 29, 31, 33, 37, 39, 42, 44, 49, 52, 55, 59, 62, 63, 64 ], [ 2, 3, 4, 5, 8, 9, 11, 14, 16, 17, 18, 20, 21, 23, 25, 27, 31, 37, 39, 42, 45, 47, 51, 52, 58, 61, 62, 64 ], [ 2, 3, 4, 5, 8, 9, 11, 13, 14, 16, 19, 20, 21, 25, 27, 31, 33, 34, 37, 39, 42, 47, 51, 52, 53, 57, 61, 62 ], [ 2, 3, 4, 5, 8, 9, 11, 13, 14, 16, 17, 18, 20, 21, 25, 29, 31, 33, 37, 39, 42, 47, 49, 52, 55, 61, 62, 63 ], [ 2, 3, 4, 5, 7, 9, 13, 14, 15, 16, 18, 19, 23, 25, 29, 31, 33, 34, 37, 41, 43, 49, 53, 56, 60, 61, 62, 64 ], [ 2, 3, 4, 5, 7, 9, 13, 14, 15, 16, 18, 19, 23, 25, 27, 31, 33, 37, 41, 45, 51, 55, 56, 58, 61, 62, 63, 64 ], [ 2, 3, 4, 5, 7, 9, 13, 14, 15, 16, 18, 19, 23, 25, 27, 31, 33, 34, 37, 41, 45, 51, 53, 56, 58, 61, 62, 64 ], [ 2, 3, 4, 5, 6, 9, 11, 13, 14, 16, 18, 19, 21, 22, 23, 27, 31, 33, 34, 37, 41, 44, 51, 52, 53, 59, 62, 64 ], [ 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 22, 25, 27, 31, 33, 34, 37, 41, 47, 51, 52, 53, 61, 62 ], [ 1, 2, 6, 7, 8, 9, 11, 13, 15, 22, 23, 25, 27, 31, 33, 34, 36, 38, 45, 47, 51, 52, 53, 56, 58, 61, 62, 64 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 29, 31, 33, 41, 44, 45, 47, 49, 52, 55, 56, 58, 59, 62, 63 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 27, 31, 33, 34, 41, 43, 44, 47, 51, 52, 53, 56, 59, 60, 62 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 23, 29, 31, 33, 41, 43, 47, 49, 52, 55, 56, 60, 62, 63, 64 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 23, 29, 31, 33, 34, 41, 43, 47, 49, 52, 53, 56, 60, 62, 64 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 23, 27, 31, 33, 41, 45, 47, 51, 52, 55, 56, 58, 62, 63, 64 ], [ 1, 2, 4, 8, 9, 10, 11, 12, 13, 15, 18, 19, 21, 23, 27, 31, 33, 34, 41, 45, 47, 51, 52, 53, 56, 58, 62, 64 ], [ 1, 2, 4, 7, 9, 11, 13, 15, 18, 19, 20, 23, 27, 31, 33, 34, 36, 38, 41, 42, 44, 51, 52, 53, 56, 59, 62, 64 ], [ 1, 2, 4, 7, 8, 9, 11, 13, 15, 18, 19, 20, 25, 27, 31, 33, 34, 36, 38, 41, 42, 47, 51, 52, 53, 56, 61, 62 ], [ 1, 2, 4, 7, 8, 9, 10, 12, 13, 18, 19, 20, 23, 25, 28, 31, 33, 41, 42, 43, 47, 50, 55, 60, 61, 62, 63, 64 ], [ 1, 2, 4, 7, 8, 9, 10, 12, 13, 18, 19, 20, 23, 25, 28, 31, 33, 34, 41, 42, 43, 47, 50, 53, 60, 61, 62, 64 ], [ 1, 2, 4, 6, 9, 13, 15, 18, 19, 20, 21, 22, 27, 31, 33, 41, 42, 43, 44, 46, 48, 51, 55, 56, 59, 60, 62, 63 ], [ 1, 2, 4, 6, 9, 13, 15, 18, 19, 20, 21, 22, 23, 27, 31, 33, 41, 42, 45, 46, 48, 51, 55, 56, 58, 62, 63, 64 ], [ 1, 2, 4, 6, 9, 13, 15, 18, 19, 20, 21, 22, 23, 27, 31, 33, 34, 41, 42, 45, 46, 48, 51, 53, 56, 58, 62, 64 ], [ 1, 2, 4, 6, 7, 9, 11, 13, 15, 19, 22, 23, 27, 31, 33, 34, 36, 38, 39, 44, 51, 52, 53, 56, 57, 59, 62, 64 ], [ 1, 2, 4, 6, 7, 9, 11, 13, 15, 17, 18, 22, 23, 29, 31, 33, 36, 38, 39, 44, 49, 52, 55, 56, 59, 62, 63, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 13, 17, 19, 22, 23, 27, 30, 31, 32, 33, 34, 40, 45, 46, 47, 48, 51, 53, 58, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 17, 18, 22, 23, 25, 27, 31, 34, 39, 45, 47, 51, 52, 53, 55, 58, 61, 62, 63, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 15, 17, 18, 22, 23, 25, 27, 31, 36, 38, 39, 45, 47, 51, 52, 56, 58, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 13, 17, 19, 22, 23, 25, 27, 31, 33, 34, 40, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 13, 17, 18, 22, 23, 25, 27, 31, 33, 35, 39, 45, 47, 51, 52, 54, 58, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 13, 15, 19, 22, 25, 27, 31, 33, 34, 36, 38, 39, 47, 51, 52, 53, 56, 57, 61, 62 ], [ 1, 2, 4, 6, 7, 8, 9, 11, 13, 15, 17, 18, 22, 25, 29, 31, 33, 36, 38, 39, 47, 49, 52, 55, 56, 61, 62, 63 ], [ 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 19, 22, 23, 25, 28, 31, 33, 39, 43, 47, 50, 55, 57, 60, 61, 62, 63, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 19, 22, 23, 25, 28, 31, 33, 34, 39, 43, 47, 50, 53, 57, 60, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 17, 22, 23, 25, 28, 31, 33, 34, 41, 45, 47, 50, 53, 57, 58, 61, 62, 64 ], [ 1, 2, 4, 6, 7, 8, 9, 10, 12, 13, 17, 18, 22, 23, 25, 28, 31, 33, 39, 45, 47, 50, 55, 58, 61, 62, 63, 64 ], [ 1, 2, 4, 5, 7, 8, 9, 11, 13, 18, 19, 21, 23, 25, 27, 31, 33, 41, 45, 47, 51, 52, 55, 58, 61, 62, 63, 64 ], [ 1, 2, 4, 5, 7, 8, 9, 11, 13, 18, 19, 21, 23, 25, 27, 31, 33, 34, 41, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 1, 2, 3, 6, 7, 8, 9, 11, 13, 14, 16, 22, 23, 25, 27, 31, 33, 34, 37, 45, 47, 51, 52, 53, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 9, 13, 14, 15, 16, 18, 19, 21, 27, 31, 33, 37, 41, 43, 44, 46, 48, 51, 55, 56, 59, 60, 62, 63 ], [ 1, 2, 3, 4, 9, 13, 14, 15, 16, 18, 19, 21, 23, 27, 31, 33, 37, 41, 45, 46, 48, 51, 55, 56, 58, 62, 63, 64 ], [ 1, 2, 3, 4, 9, 13, 14, 15, 16, 18, 19, 21, 23, 27, 31, 33, 34, 37, 41, 45, 46, 48, 51, 53, 56, 58, 62, 64 ], [ 1, 2, 3, 4, 7, 9, 11, 13, 14, 16, 18, 19, 20, 23, 27, 31, 33, 34, 37, 41, 42, 44, 51, 52, 53, 59, 62, 64 ], [ 1, 2, 3, 4, 7, 8, 9, 11, 13, 14, 16, 18, 19, 20, 25, 27, 31, 33, 34, 37, 41, 42, 47, 51, 52, 53, 61, 62 ], [ 1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 16, 19, 22, 23, 27, 31, 33, 34, 37, 39, 44, 51, 52, 53, 57, 59, 62, 64 ], [ 1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 16, 17, 18, 22, 23, 29, 31, 33, 37, 39, 44, 49, 52, 55, 59, 62, 63, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 14, 16, 17, 18, 22, 23, 25, 27, 31, 37, 39, 45, 47, 51, 52, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 16, 19, 22, 25, 27, 31, 33, 34, 37, 39, 47, 51, 52, 53, 57, 61, 62 ], [ 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 22, 25, 29, 31, 33, 37, 39, 47, 49, 52, 55, 61, 62, 63 ] ]; ########################### ########################### ###########################