Best Known (40, 40+∞, s)-Nets in Base 7
(40, 40+∞, 41)-Net over F7 — Constructive and digital
Digital (40, m, 41)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (40, 40)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 40)-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, 40)-sequence over F49, using
(40, 40+∞, 96)-Net over F7 — Digital
Digital (40, m, 96)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (40, 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
(40, 40+∞, 260)-Net in Base 7 — Upper bound on s
There is no (40, m, 261)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (40, 779, 261)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7779, 261, S7, 3, 739), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 4 804201 222906 608843 595495 974443 439468 666934 066804 430507 559939 628609 176518 535405 005739 677651 050148 007901 872631 123724 004034 686927 839778 504767 195830 316025 275201 775111 268345 910402 485984 818296 082640 794545 636198 501573 655634 342679 467885 185341 239914 713977 746798 452431 045493 047239 156360 645004 781036 935641 991246 147939 924802 876931 049896 163196 443995 841160 806268 173795 366856 326744 256011 432991 448919 223161 860798 650729 790464 430207 464520 576483 995597 243290 450619 608295 953927 755136 344776 244305 400076 718792 380111 763150 187962 681729 004803 872970 539235 407240 969451 588161 897398 703515 711943 894825 139838 268962 336026 498754 506027 213532 501341 803706 006034 382310 809158 596442 112517 206747 776032 / 185 > 7779 [i]
- extracting embedded OOA [i] would yield OOA(7779, 261, S7, 3, 739), but