Lower bounds for A179183 ======================== Pontus von Brömssen, March 2024 Below are some lower bounds that improve those for A179183(n) = a(n) = E_2(n,3) given by Sheridan Houghten's table available at https://www.cosc.brocku.ca/~houghten/binaryedit.html (as of March 2024), together with codes attaining these bounds, i.e., sets of binary strings of length n whose pairwise edit distances are at least 3. The codes are given as sets of decimal numbers in the range 0..2^n-1, to be interpreted as binary numbers (padded with leading 0's to make them of length n). These codes were found by simulated annealing search, with the objective function given by the cardinality of a set of binary strings minus a penalty factor times the number of pairs in the set at distance less than 3. a(10) >= 58: {10, 19, 54, 56, 80, 93, 103, 105, 143, 153, 165, 195, 200, 242, 255, 268, 277, 291, 319, 321, 347, 366, 372, 384, 414, 426, 433, 454, 493, 504, 513, 571, 578, 606, 620, 625, 663, 666, 678, 681, 688, 709, 783, 790, 793, 805, 808, 836, 875, 882, 893, 899, 956, 970, 977, 991, 992, 999} a(11) >= 104: {7, 30, 45, 65, 76, 112, 127, 147, 162, 218, 231, 249, 260, 315, 316, 343, 344, 358, 361, 392, 399, 406, 421, 451, 480, 510, 523, 528, 567, 570, 598, 601, 611, 616, 652, 661, 673, 735, 749, 754, 782, 787, 802, 832, 884, 922, 939, 944, 966, 969, 988, 1015, 1032, 1059, 1078, 1081, 1103, 1108, 1125, 1153, 1166, 1200, 1215, 1218, 1245, 1260, 1267, 1293, 1298, 1312, 1374, 1386, 1435, 1436, 1446, 1449, 1476, 1519, 1525, 1538, 1567, 1596, 1626, 1638, 1649, 1686, 1689, 1703, 1704, 1733, 1771, 1790, 1813, 1816, 1839, 1859, 1900, 1906, 1920, 1971, 1981, 1994, 2001, 2040} a(12) >= 179: {23, 24, 42, 70, 81, 132, 175, 185, 244, 257, 300, 309, 333, 354, 383, 392, 403, 478, 481, 504, 541, 563, 587, 624, 647, 680, 702, 705, 730, 742, 778, 822, 832, 855, 860, 873, 921, 933, 944, 975, 1002, 1011, 1021, 1033, 1063, 1076, 1100, 1120, 1147, 1166, 1170, 1185, 1239, 1240, 1258, 1277, 1296, 1311, 1338, 1366, 1369, 1381, 1429, 1442, 1475, 1516, 1538, 1566, 1593, 1621, 1635, 1660, 1677, 1681, 1700, 1775, 1778, 1796, 1811, 1832, 1839, 1882, 1905, 1948, 1975, 1990, 1993, 2000, 2046, 2053, 2078, 2098, 2122, 2173, 2187, 2192, 2231, 2236, 2262, 2265, 2277, 2280, 2310, 2351, 2361, 2389, 2404, 2445, 2467, 2490, 2498, 2542, 2560, 2606, 2613, 2616, 2637, 2644, 2658, 2707, 2783, 2795, 2801, 2828, 2834, 2849, 2907, 2919, 2942, 2974, 2982, 3013, 3016, 3060, 3101, 3116, 3121, 3155, 3198, 3207, 3208, 3259, 3265, 3302, 3312, 3338, 3360, 3383, 3407, 3420, 3433, 3478, 3481, 3493, 3524, 3546, 3583, 3595, 3638, 3662, 3673, 3685, 3732, 3747, 3773, 3778, 3831, 3861, 3864, 3899, 3905, 3954, 3968, 3983, 4012, 4017, 4051, 4061, 4088}