Best Known (107, ∞, s)-Nets in Base 2
(107, ∞, 56)-Net over F2 — Constructive and digital
Digital (107, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (107, 55)-sequence over F2, using
- t-expansion [i] based on digital (105, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 7 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (105, 55)-sequence over F2, using
(107, ∞, 65)-Net over F2 — Digital
Digital (107, m, 65)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (107, 64)-sequence over F2, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
(107, ∞, 117)-Net in Base 2 — Upper bound on s
There is no (107, m, 118)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (107, 816, 118)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2816, 118, S2, 7, 709), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 241 221236 618014 199597 765636 160103 009748 510742 539035 037174 250491 877375 905318 524217 162454 176555 407636 462967 017350 599484 946915 696214 872810 423868 494482 039545 409401 292520 416328 812873 096836 797125 187446 399873 116209 768319 804590 844587 999666 682099 158977 871872 / 355 > 2816 [i]
- extracting embedded OOA [i] would yield OOA(2816, 118, S2, 7, 709), but