### 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 [64,143]. 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 [64,143]. # 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(64,143));' can be # used to create a list e of the elements of the group [64,143]. # In our list of (64,28,12) DSs in group [64,143], we have [1, # 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 30, # 34, 35, 36, 40, 43, 49, 50, 51, 55, 56, 57, 60, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[7], e[9], e[10], e[11], e[13], e[16], e[18], e[21], e[23], e[30], e[34], # e[35], e[36], e[40], e[43], e[49], e[50], e[51], e[55], e[56], e[57], e[60], e[64]] # where e is the list of the elements of the group [64,143] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,143));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 143] ### The number of inequivalent (64,28,12) DSs in group [64,143] is 3134 ### The list of all inequivalent (64,28,12) DSs in group [64,143] is DS143:=[ [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 30, 34, 35, 36, 40, 43, 49, 50, 51, 55, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 30, 34, 35, 36, 40, 43, 44, 49, 50, 51, 55, 56, 57, 58 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 28, 30, 34, 35, 36, 40, 43, 50, 55, 56, 57, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 28, 30, 34, 35, 36, 40, 43, 44, 50, 55, 56, 57, 58, 62 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 28, 29, 30, 34, 36, 40, 43, 51, 54, 56, 57, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 21, 23, 28, 29, 30, 34, 36, 40, 43, 44, 51, 54, 56, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 19, 21, 23, 30, 34, 36, 41, 43, 49, 50, 51, 54, 56, 60, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 19, 21, 23, 30, 34, 36, 41, 43, 44, 49, 50, 51, 54, 56, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 19, 21, 23, 28, 30, 34, 36, 41, 43, 50, 54, 56, 60, 62, 63, 64 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 18, 19, 21, 23, 28, 30, 34, 36, 41, 43, 44, 50, 54, 56, 58, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 34, 35, 36, 37, 39, 43, 44, 49, 50, 51, 52, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 28, 29, 34, 36, 37, 39, 43, 44, 51, 52, 54, 57, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 28, 29, 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, 35, 36, 37, 41, 43, 44, 49, 50, 51, 52, 53, 58, 63 ], [ 1, 2, 3, 4, 5, 7, 9, 10, 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number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 1801 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS143[7], DS143[8], DS143[9], DS143[10], DS143[11], DS143[12], DS143[13], DS143[15], DS143[16], DS143[17], DS143[23], DS143[24], DS143[25], DS143[26], DS143[28], DS143[31], DS143[35], DS143[37], DS143[40], DS143[45], DS143[54], DS143[55], DS143[56], DS143[63], DS143[67], DS143[69], DS143[73], DS143[77], DS143[80], DS143[81], DS143[82], DS143[84], DS143[86], DS143[90], DS143[91], DS143[93], DS143[94], DS143[98], DS143[100], DS143[101], DS143[102], DS143[103], DS143[105], DS143[106], DS143[107], DS143[108], DS143[110], DS143[111], DS143[114], DS143[115], DS143[116], DS143[117], DS143[118], DS143[122], DS143[123], DS143[125], DS143[128], DS143[129], DS143[130], DS143[137], DS143[140], DS143[142], DS143[155], DS143[156], DS143[157], DS143[158], DS143[167], DS143[168], DS143[169], DS143[170], DS143[172], DS143[174], DS143[177], DS143[183], DS143[189], DS143[192], DS143[193], DS143[198], DS143[201], DS143[202], DS143[204], DS143[205], DS143[210], DS143[211], DS143[213], DS143[214], DS143[215], DS143[216], DS143[219], DS143[220], DS143[223], DS143[226], DS143[227], DS143[228], DS143[229], DS143[232], DS143[233], DS143[235], DS143[240], DS143[241], DS143[249], DS143[250], DS143[251], DS143[252], DS143[255], DS143[258], DS143[259], DS143[260], DS143[261], DS143[265], DS143[266], DS143[268], DS143[269], DS143[270], DS143[271], DS143[272], DS143[273], DS143[274], DS143[277], DS143[291], DS143[292], DS143[294], DS143[300], DS143[303], DS143[309], DS143[310], DS143[315], DS143[331], DS143[340], DS143[349], DS143[350], DS143[358], DS143[364], DS143[366], DS143[367], DS143[379], DS143[383], DS143[390], DS143[391], DS143[396], DS143[397], DS143[398], DS143[404], DS143[407], DS143[409], DS143[410], DS143[418], DS143[422], DS143[424], DS143[425], DS143[430], DS143[435], DS143[448], DS143[449], DS143[456], DS143[457], DS143[459], DS143[460], DS143[461], DS143[468], DS143[479], DS143[480], DS143[485], DS143[495], DS143[498], DS143[499], DS143[500], DS143[501], DS143[502], DS143[504], DS143[508], DS143[509], DS143[511], DS143[519], DS143[520], DS143[525], DS143[526], DS143[527], DS143[528], DS143[530], DS143[531], DS143[532], DS143[537], DS143[541], DS143[543], DS143[544], DS143[551], DS143[552], DS143[553], DS143[554], DS143[557], DS143[559], DS143[560], DS143[567], DS143[568], DS143[569], DS143[570], DS143[571], DS143[575], DS143[576], DS143[577], DS143[579], DS143[582], DS143[596], DS143[600], DS143[603], DS143[604], DS143[605], DS143[606], DS143[607], DS143[611], DS143[613], DS143[614], DS143[616], DS143[620], DS143[623], DS143[624], DS143[626], DS143[635], DS143[636], DS143[638], DS143[639], DS143[640], DS143[642], DS143[643], DS143[644], DS143[648], DS143[651], DS143[656], DS143[657], DS143[660], DS143[661], DS143[664], DS143[675], DS143[676], DS143[677], DS143[678], DS143[679], DS143[680], DS143[681], DS143[682], DS143[683], DS143[684], DS143[685], DS143[686], DS143[687], DS143[688], DS143[689], DS143[691], DS143[693], DS143[695], DS143[698], DS143[704], DS143[709], DS143[710], DS143[715], DS143[718], DS143[722], DS143[723], DS143[724], DS143[727], DS143[728], DS143[729], DS143[731], DS143[734], DS143[736], DS143[737], DS143[738], DS143[739], DS143[743], DS143[746], DS143[747], DS143[752], DS143[753], DS143[754], DS143[759], DS143[761], DS143[766], DS143[772], DS143[776], DS143[777], DS143[787], DS143[788], DS143[789], DS143[790], DS143[793], DS143[794], DS143[795], DS143[797], DS143[800], DS143[801], DS143[802], DS143[803], DS143[804], DS143[806], DS143[810], DS143[812], DS143[816], DS143[817], DS143[818], DS143[819], DS143[820], DS143[821], DS143[823], DS143[824], DS143[826], DS143[827], DS143[828], DS143[830], DS143[831], DS143[833], DS143[836], DS143[841], DS143[842], DS143[843], DS143[844], DS143[845], DS143[846], DS143[848], DS143[849], DS143[850], DS143[851], DS143[852], DS143[853], DS143[854], DS143[855], DS143[856], DS143[859], DS143[860], DS143[864], DS143[872], DS143[876], DS143[878], DS143[879], DS143[880], DS143[881], DS143[882], DS143[883], DS143[887], DS143[888], DS143[890], DS143[891], DS143[892], DS143[893], DS143[894], DS143[896], DS143[897], DS143[900], DS143[903], DS143[906], DS143[907], DS143[908], DS143[912], DS143[913], DS143[914], DS143[919], DS143[920], DS143[921], DS143[922], DS143[929], DS143[945], DS143[948], DS143[949], DS143[950], DS143[951], DS143[952], DS143[953], DS143[954], DS143[956], DS143[963], DS143[973], DS143[974], DS143[976], DS143[977], DS143[984], DS143[987], DS143[992], DS143[993], DS143[995], DS143[998], DS143[999], DS143[1001], DS143[1006], DS143[1008], DS143[1009], DS143[1011], DS143[1012], DS143[1014], DS143[1016], DS143[1018], DS143[1021], DS143[1022], DS143[1024], DS143[1025], DS143[1026], DS143[1031], DS143[1033], DS143[1040], DS143[1041], DS143[1042], DS143[1043], DS143[1044], DS143[1045], DS143[1046], DS143[1047], DS143[1048], DS143[1049], DS143[1050], DS143[1051], DS143[1052], DS143[1053], DS143[1054], DS143[1055], DS143[1056], DS143[1057], DS143[1058], DS143[1059], DS143[1060], DS143[1061], DS143[1062], DS143[1066], DS143[1068], DS143[1071], DS143[1072], DS143[1080], DS143[1082], DS143[1083], DS143[1087], DS143[1091], DS143[1095], DS143[1096], DS143[1097], DS143[1098], DS143[1100], DS143[1105], DS143[1120], DS143[1121], DS143[1122], DS143[1130], DS143[1132], DS143[1138], DS143[1144], DS143[1150], DS143[1153], DS143[1156], DS143[1157], DS143[1158], DS143[1159], DS143[1160], DS143[1161], DS143[1163], DS143[1164], DS143[1165], DS143[1166], DS143[1167], DS143[1169], DS143[1170], DS143[1171], DS143[1172], DS143[1174], DS143[1175], DS143[1176], DS143[1180], DS143[1181], DS143[1182], DS143[1183], DS143[1184], DS143[1185], DS143[1188], DS143[1193], DS143[1194], DS143[1195], DS143[1197], DS143[1199], DS143[1200], DS143[1201], DS143[1205], DS143[1206], DS143[1207], DS143[1208], DS143[1209], DS143[1210], DS143[1211], DS143[1212], DS143[1213], DS143[1214], DS143[1215], DS143[1216], DS143[1217], DS143[1218], DS143[1219], DS143[1220], DS143[1221], DS143[1222], DS143[1229], DS143[1233], DS143[1234], DS143[1235], DS143[1240], DS143[1247], DS143[1248], DS143[1250], DS143[1251], DS143[1252], DS143[1253], DS143[1254], DS143[1255], DS143[1256], DS143[1257], DS143[1259], DS143[1260], DS143[1261], DS143[1262], DS143[1264], DS143[1265], DS143[1266], DS143[1267], DS143[1268], DS143[1269], DS143[1271], DS143[1272], DS143[1273], DS143[1278], DS143[1291], DS143[1299], DS143[1303], DS143[1305], DS143[1307], DS143[1308], DS143[1309], DS143[1310], DS143[1311], DS143[1312], DS143[1313], DS143[1317], DS143[1318], DS143[1321], DS143[1324], DS143[1325], DS143[1326], DS143[1327], DS143[1329], DS143[1330], DS143[1331], DS143[1332], DS143[1340], DS143[1343], DS143[1350], DS143[1351], DS143[1352], DS143[1353], DS143[1357], DS143[1358], DS143[1359], DS143[1360], DS143[1362], DS143[1365], DS143[1368], DS143[1371], DS143[1372], DS143[1373], DS143[1374], DS143[1377], DS143[1379], DS143[1380], DS143[1381], DS143[1382], DS143[1384], DS143[1389], DS143[1392], DS143[1393], DS143[1394], DS143[1396], DS143[1399], DS143[1400], DS143[1401], DS143[1402], DS143[1403], DS143[1404], DS143[1405], DS143[1406], DS143[1407], DS143[1410], DS143[1411], DS143[1414], DS143[1415], DS143[1416], DS143[1420], DS143[1422], DS143[1423], DS143[1424], DS143[1428], DS143[1429], DS143[1430], DS143[1431], DS143[1437], DS143[1438], DS143[1439], DS143[1440], DS143[1444], DS143[1446], DS143[1447], DS143[1448], DS143[1449], DS143[1450], DS143[1451], DS143[1452], DS143[1454], DS143[1457], DS143[1458], DS143[1461], DS143[1463], DS143[1465], DS143[1478], DS143[1480], DS143[1482], DS143[1487], DS143[1489], DS143[1491], DS143[1494], DS143[1495], DS143[1496], DS143[1497], DS143[1499], DS143[1503], DS143[1505], DS143[1508], DS143[1509], DS143[1511], DS143[1516], DS143[1517], DS143[1521], DS143[1524], DS143[1526], DS143[1527], DS143[1531], DS143[1534], DS143[1536], DS143[1537], DS143[1538], DS143[1539], DS143[1544], DS143[1546], DS143[1547], DS143[1548], DS143[1549], DS143[1550], DS143[1551], DS143[1565], DS143[1566], DS143[1567], DS143[1568], DS143[1569], DS143[1573], DS143[1574], DS143[1575], DS143[1588], DS143[1590], DS143[1591], DS143[1593], DS143[1595], DS143[1602], DS143[1606], DS143[1611], DS143[1612], DS143[1615], DS143[1616], DS143[1617], DS143[1618], DS143[1621], DS143[1622], DS143[1624], DS143[1627], DS143[1628], DS143[1629], DS143[1630], DS143[1633], DS143[1634], DS143[1638], DS143[1639], DS143[1640], DS143[1641], DS143[1642], DS143[1643], DS143[1644], DS143[1645], DS143[1648], DS143[1657], DS143[1658], DS143[1659], DS143[1660], DS143[1661], DS143[1668], DS143[1669], DS143[1672], DS143[1673], DS143[1674], DS143[1677], DS143[1678], DS143[1679], DS143[1680], DS143[1681], DS143[1682], DS143[1685], DS143[1686], DS143[1687], DS143[1692], DS143[1699], DS143[1700], DS143[1702], DS143[1706], DS143[1714], DS143[1717], DS143[1722], DS143[1723], DS143[1724], DS143[1725], DS143[1726], DS143[1729], DS143[1733], DS143[1734], DS143[1735], DS143[1736], DS143[1738], DS143[1739], DS143[1741], DS143[1742], DS143[1743], DS143[1744], DS143[1748], DS143[1755], DS143[1759], DS143[1761], DS143[1764], DS143[1765], DS143[1766], DS143[1767], DS143[1771], DS143[1773], DS143[1778], DS143[1783], DS143[1786], DS143[1790], DS143[1791], DS143[1792], DS143[1793], DS143[1797], DS143[1800], DS143[1801], DS143[1802], DS143[1803], DS143[1804], DS143[1805], DS143[1806], DS143[1807], DS143[1808], DS143[1810], DS143[1812], DS143[1813], DS143[1814], DS143[1819], DS143[1822], DS143[1824], DS143[1825], DS143[1827], DS143[1828], DS143[1829], DS143[1831], DS143[1834], DS143[1839], DS143[1842], DS143[1844], DS143[1845], DS143[1850], DS143[1852], DS143[1853], DS143[1855], DS143[1857], DS143[1858], DS143[1859], DS143[1862], DS143[1868], DS143[1874], DS143[1880], DS143[1886], DS143[1889], DS143[1890], DS143[1895], DS143[1896], DS143[1897], DS143[1898], DS143[1899], DS143[1902], DS143[1903], DS143[1904], DS143[1906], DS143[1907], DS143[1908], DS143[1909], DS143[1910], DS143[1912], DS143[1916], DS143[1917], DS143[1918], DS143[1919], DS143[1921], DS143[1922], DS143[1924], DS143[1927], DS143[1928], DS143[1929], DS143[1930], DS143[1946], DS143[1952], DS143[1954], DS143[1985], DS143[1987], DS143[1988], DS143[2005], DS143[2006], DS143[2011], DS143[2015], DS143[2016], DS143[2017], DS143[2018], DS143[2023], DS143[2024], DS143[2025], DS143[2026], DS143[2027], DS143[2028], DS143[2029], DS143[2030], DS143[2031], DS143[2032], DS143[2033], DS143[2035], DS143[2036], DS143[2037], DS143[2039], DS143[2040], DS143[2043], DS143[2044], DS143[2047], DS143[2049], DS143[2050], DS143[2051], DS143[2052], DS143[2054], DS143[2055], DS143[2057], DS143[2059], DS143[2062], DS143[2065], DS143[2067], DS143[2068], DS143[2069], DS143[2070], DS143[2072], DS143[2073], DS143[2074], DS143[2075], DS143[2076], DS143[2077], DS143[2079], DS143[2081], DS143[2082], DS143[2083], DS143[2084], DS143[2086], DS143[2089], DS143[2090], DS143[2091], DS143[2092], DS143[2093], DS143[2094], DS143[2095], DS143[2096], DS143[2097], DS143[2098], DS143[2101], DS143[2102], DS143[2103], DS143[2104], DS143[2105], DS143[2106], DS143[2107], DS143[2108], DS143[2109], DS143[2110], DS143[2111], DS143[2112], DS143[2113], DS143[2114], DS143[2115], DS143[2116], DS143[2117], DS143[2118], DS143[2119], DS143[2120], DS143[2121], DS143[2122], DS143[2123], DS143[2124], DS143[2125], DS143[2126], DS143[2127], DS143[2128], DS143[2129], DS143[2130], DS143[2131], DS143[2132], DS143[2133], DS143[2134], DS143[2135], DS143[2136], DS143[2137], DS143[2139], DS143[2140], DS143[2141], DS143[2142], DS143[2143], DS143[2144], DS143[2145], DS143[2146], DS143[2147], DS143[2148], DS143[2149], DS143[2150], DS143[2151], DS143[2153], DS143[2158], DS143[2159], DS143[2160], DS143[2161], DS143[2162], DS143[2163], DS143[2164], DS143[2165], DS143[2166], DS143[2167], DS143[2168], DS143[2169], DS143[2170], DS143[2171], DS143[2172], DS143[2173], DS143[2174], DS143[2175], DS143[2176], DS143[2177], DS143[2178], DS143[2179], DS143[2180], DS143[2181], DS143[2182], DS143[2183], DS143[2184], DS143[2185], DS143[2186], DS143[2188], DS143[2190], DS143[2191], DS143[2192], DS143[2193], DS143[2194], DS143[2195], DS143[2198], DS143[2199], DS143[2200], DS143[2201], DS143[2202], DS143[2203], DS143[2204], DS143[2205], DS143[2206], DS143[2207], DS143[2214], DS143[2215], DS143[2216], DS143[2217], DS143[2218], DS143[2224], DS143[2225], DS143[2226], DS143[2227], DS143[2228], DS143[2229], DS143[2230], DS143[2233], DS143[2234], DS143[2235], DS143[2236], DS143[2237], DS143[2238], DS143[2239], DS143[2240], DS143[2241], DS143[2242], DS143[2243], DS143[2244], DS143[2245], DS143[2246], DS143[2247], DS143[2248], DS143[2249], DS143[2250], DS143[2251], DS143[2252], DS143[2253], DS143[2254], DS143[2255], DS143[2256], DS143[2257], DS143[2258], DS143[2259], DS143[2260], 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DS143[3121], DS143[3122], DS143[3123], DS143[3124], DS143[3125], DS143[3126], DS143[3127], DS143[3128], DS143[3129], DS143[3130], DS143[3131], DS143[3132], DS143[3133], DS143[3134]. # Note that there is no reversible (64,28,12) difference sets in # group [64,143]. ########################### ########################### ###########################