Best Known (45, 45+∞, s)-Nets in Base 7
(45, 45+∞, 46)-Net over F7 — Constructive and digital
Digital (45, m, 46)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 45)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 45)-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, 45)-sequence over F49, using
(45, 45+∞, 105)-Net over F7 — Digital
Digital (45, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (45, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(45, 45+∞, 290)-Net in Base 7 — Upper bound on s
There is no (45, m, 291)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (45, 869, 291)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7869, 291, S7, 3, 824), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 70482 073028 799291 209260 152681 771006 552661 617073 157573 567459 727521 438089 754986 082626 276408 137211 378701 811708 067753 226218 939519 626060 049351 974685 032074 067892 807444 265631 676300 566079 676213 781229 496465 430731 880513 483677 830984 139203 466274 592730 776822 794265 442049 768751 491555 381719 415805 903234 086684 094507 674771 753208 957298 663857 394288 451180 948275 694100 989895 322474 168725 249265 395865 266496 707158 929850 517583 559229 673760 332357 589266 043926 409546 352114 828533 728829 499209 670586 316929 625884 791335 482864 543783 363577 851017 989879 668915 965591 044288 757413 309288 148670 750099 427082 348421 859218 383679 059004 889903 430386 286410 816688 705127 505862 712060 825045 640728 846499 230772 464409 386143 339598 189203 012910 836649 807737 544796 087100 283498 347845 103884 095968 529209 / 275 > 7869 [i]
- extracting embedded OOA [i] would yield OOA(7869, 291, S7, 3, 824), but