Best Known (34, 34+∞, s)-Nets in Base 7
(34, 34+∞, 35)-Net over F7 — Constructive and digital
Digital (34, m, 35)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (34, 34)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 34)-sequence over F49, using
- s-reduction based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- s-reduction based on digital (0, 49)-sequence over F49, using
- base reduction for sequences [i] based on digital (0, 34)-sequence over F49, using
(34, 34+∞, 96)-Net over F7 — Digital
Digital (34, m, 96)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (34, 95)-sequence over F7, using
- t-expansion [i] based on digital (33, 95)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 33 and N(F) ≥ 96, using
- t-expansion [i] based on digital (33, 95)-sequence over F7, using
(34, 34+∞, 224)-Net in Base 7 — Upper bound on s
There is no (34, m, 225)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (34, 671, 225)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7671, 225, S7, 3, 637), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 539458 067502 845001 142950 903867 991867 691965 310836 747861 288962 935051 549983 485712 092104 045125 657453 378150 168591 262875 211468 913344 946602 506038 852940 515999 040148 470188 965128 158693 973359 368566 003539 295286 202351 321186 167272 908225 468782 115454 279476 110242 230371 746226 388814 001811 404189 782727 338946 275920 343634 058289 736518 724091 486759 738413 834745 550066 884968 731970 975590 818404 620561 340791 382195 332488 360155 036483 533451 818948 598537 618804 778686 021528 589082 659628 096782 801735 787122 402412 294907 561056 765109 024891 512598 359377 156897 570336 045880 623196 929918 703068 169557 037138 594640 774467 / 319 > 7671 [i]
- extracting embedded OOA [i] would yield OOA(7671, 225, S7, 3, 637), but