Best Known (68, 68+∞, s)-Nets in Base 3
(68, 68+∞, 48)-Net over F3 — Constructive and digital
Digital (68, m, 48)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 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
(68, 68+∞, 72)-Net over F3 — Digital
Digital (68, m, 72)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 71)-sequence over F3, using
- t-expansion [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
- t-expansion [i] based on digital (67, 71)-sequence over F3, using
(68, 68+∞, 148)-Net in Base 3 — Upper bound on s
There is no (68, m, 149)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (68, 739, 149)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3739, 149, S3, 5, 671), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5 675122 495595 460141 465981 106555 118127 792505 745324 384070 514479 548947 068354 777023 135830 757612 072347 370567 041513 140819 622780 584331 071222 588705 422541 762520 369795 916489 414571 366800 473985 524843 331498 918986 680515 143655 303206 406935 998417 160695 279156 081303 019220 724778 430539 650273 849328 436615 931713 408266 793526 488960 400109 927360 546571 312924 986045 862318 050091 168715 / 112 > 3739 [i]
- extracting embedded OOA [i] would yield OOA(3739, 149, S3, 5, 671), but