Best Known (105, 105+∞, s)-Nets in Base 3
(105, 105+∞, 72)-Net over F3 — Constructive and digital
Digital (105, m, 72)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 71)-sequence over F3, using
- base reduction for sequences [i] based on digital (17, 71)-sequence over F9, using
- s-reduction based on digital (17, 73)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 17 and N(F) ≥ 74, using
- s-reduction based on digital (17, 73)-sequence over F9, using
- base reduction for sequences [i] based on digital (17, 71)-sequence over F9, using
(105, 105+∞, 104)-Net over F3 — Digital
Digital (105, m, 104)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 103)-sequence over F3, using
- t-expansion [i] based on digital (102, 103)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 102 and N(F) ≥ 104, using
- t-expansion [i] based on digital (102, 103)-sequence over F3, using
(105, 105+∞, 223)-Net in Base 3 — Upper bound on s
There is no (105, m, 224)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (105, 1113, 224)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(31113, 224, S3, 5, 1008), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 280767 527805 988499 806015 976304 145579 030571 529793 023810 717554 799761 230281 344004 427605 526613 937155 481036 658391 803020 823797 977606 237429 371764 983314 796398 710466 533933 005435 534474 263639 396501 920703 050052 691736 484138 985446 957359 550208 607959 310164 090141 335744 827954 568712 562688 132680 710730 070590 466020 342758 850910 404696 984466 805051 235125 939474 931411 118161 608875 416764 751278 570179 431379 333698 619187 731827 497780 553631 398465 839390 459127 874279 396044 000197 963630 851624 829351 779450 936733 102886 194714 249406 783648 426886 712075 313531 087858 484571 920817 / 1009 > 31113 [i]
- extracting embedded OOA [i] would yield OOA(31113, 224, S3, 5, 1008), but