Best Known (67, ∞, s)-Nets in Base 3
(67, ∞, 48)-Net over F3 — Constructive and digital
Digital (67, m, 48)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (67, 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
(67, ∞, 72)-Net over F3 — Digital
Digital (67, m, 72)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 67 and N(F) ≥ 72, using
(67, ∞, 146)-Net in Base 3 — Upper bound on s
There is no (67, m, 147)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (67, 729, 147)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3729, 147, S3, 5, 662), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 192 880214 178438 468246 605303 486413 606839 395122 097824 748063 672840 819354 223200 966787 084104 956095 849453 473767 149588 095276 830773 511927 467109 234619 052166 832038 104019 011495 378796 188430 056602 637950 535082 835952 610941 678908 776879 566225 307841 209880 775162 142858 396764 687016 018495 244999 933377 954981 412701 877124 482380 009060 439224 800667 482149 782228 895738 365104 442953 / 221 > 3729 [i]
- extracting embedded OOA [i] would yield OOA(3729, 147, S3, 5, 662), but