Best Known (36, ∞, s)-Nets in Base 7
(36, ∞, 37)-Net over F7 — Constructive and digital
Digital (36, m, 37)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (36, 36)-sequence over F7, using
- base reduction for sequences [i] based on digital (0, 36)-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, 36)-sequence over F49, using
(36, ∞, 96)-Net over F7 — Digital
Digital (36, m, 96)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (36, 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
(36, ∞, 236)-Net in Base 7 — Upper bound on s
There is no (36, m, 237)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (36, 707, 237)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7707, 237, S7, 3, 671), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 33551 116362 679030 638211 754453 352241 648893 587517 129324 668148 409592 402660 039400 705621 134716 379072 903557 074025 996955 696978 800825 650234 648040 180201 043726 181323 052337 727612 545302 496869 602040 919324 524723 626201 950079 963738 941308 253244 468536 870244 907499 384147 098822 526314 104536 294412 801990 185079 045469 466915 704001 670137 699992 875984 853355 475590 610245 044571 001862 445533 145615 698447 182871 778055 011502 118206 031910 040763 590803 068548 716052 181107 863804 091938 608302 833633 480064 526678 995539 786724 159532 002888 088495 999010 864557 861229 880340 673008 126550 155808 489684 350821 866424 366243 523284 604878 839484 634516 928185 718973 / 8 > 7707 [i]
- extracting embedded OOA [i] would yield OOA(7707, 237, S7, 3, 671), but