Best Known (218, ∞, s)-Nets in Base 3
(218, ∞, 112)-Net over F3 — Constructive and digital
Digital (218, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (218, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(218, ∞, 163)-Net over F3 — Digital
Digital (218, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (218, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(218, ∞, 450)-Net in Base 3 — Upper bound on s
There is no (218, m, 451)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (218, 2699, 451)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32699, 451, S3, 6, 2481), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 7 326178 106902 765999 351713 568800 013879 082722 970041 853584 827292 475756 940251 142015 892529 463728 394438 309448 859896 069768 339867 863476 445606 931022 335994 201566 230657 047873 651449 977249 532106 651425 281698 422574 825734 110023 881724 669220 388715 044675 836801 529851 150594 570518 274555 302151 893450 185994 269401 499511 996731 303627 942733 732696 110223 202218 394734 529552 152731 048847 613916 657022 564210 420268 216940 111146 449878 454534 019660 589842 584345 664197 083560 716971 219444 871456 773888 875943 344241 356991 604605 132740 340468 530141 537361 809848 365074 748580 401877 394017 840545 690338 562996 373505 283849 547472 164754 168977 252164 959110 911617 714896 038386 892570 711666 209520 299967 604352 647582 765440 868574 048187 359510 315294 961124 961115 947725 480994 272591 669108 251366 702563 292742 462002 106938 051613 058419 866350 420644 680269 175210 326622 383055 909315 911984 046479 339449 339869 303146 348592 224677 289006 340882 129625 032059 822252 252055 998877 859185 496636 447375 482830 721042 670253 941283 200352 921245 604602 878839 904701 158857 553289 624259 638719 826043 768099 239976 178043 601915 034730 979614 679576 355390 512628 531856 258972 841762 198542 363816 777663 759306 903983 831041 188250 506227 350328 427486 162496 959044 330121 917242 534596 219047 735194 127612 604073 643893 533053 568122 565140 945069 033883 914673 319565 523505 992918 799911 323329 692273 760076 408009 749840 769204 986319 945598 796434 / 1241 > 32699 [i]
- extracting embedded OOA [i] would yield OOA(32699, 451, S3, 6, 2481), but