Best Known (180, ∞, s)-Nets in Base 2
(180, ∞, 66)-Net over F2 — Constructive and digital
Digital (180, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (180, 65)-sequence over F2, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
(180, ∞, 81)-Net over F2 — Digital
Digital (180, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (180, 80)-sequence over F2, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
(180, ∞, 190)-Net in Base 2 — Upper bound on s
There is no (180, m, 191)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (180, 1709, 191)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21709, 191, S2, 9, 1529), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 414030 150555 905714 387003 055721 500554 131208 835145 885405 224815 998726 069859 719346 899170 609971 201530 630910 334716 843480 898029 043544 932117 900514 613003 557529 758278 955216 416510 897160 883424 968852 399993 191732 830737 785263 661667 458852 726429 821338 145697 211787 387403 920096 875476 951117 456423 654697 109828 989258 972226 391974 206826 713704 674629 336129 734928 273438 473711 700362 537951 169110 547189 994461 868982 404395 595207 941785 627301 971974 791718 972793 255895 459874 976086 482243 018030 145261 034571 219784 687722 937857 634985 356169 418022 963367 641088 / 45 > 21709 [i]
- extracting embedded OOA [i] would yield OOA(21709, 191, S2, 9, 1529), but