Best Known (55, ∞, s)-Nets in Base 3
(55, ∞, 48)-Net over F3 — Constructive and digital
Digital (55, m, 48)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (55, 47)-sequence over F3, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- t-expansion [i] based on digital (45, 47)-sequence over F3, using
(55, ∞, 64)-Net over F3 — Digital
Digital (55, m, 64)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (55, 63)-sequence over F3, using
- t-expansion [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- t-expansion [i] based on digital (49, 63)-sequence over F3, using
(55, ∞, 122)-Net in Base 3 — Upper bound on s
There is no (55, m, 123)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (55, 486, 123)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3486, 123, S3, 4, 431), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 38010 168784 148440 897678 060509 636712 173990 031114 566729 410483 358592 310132 254237 791928 191995 665223 200049 287565 633954 980531 708292 413683 857313 462613 317080 418068 546985 952917 369570 501215 189599 353260 715230 007106 036180 224801 800289 726046 515645 / 4 > 3486 [i]
- extracting embedded OOA [i] would yield OOA(3486, 123, S3, 4, 431), but