Best Known (108, 108+∞, s)-Nets in Base 2
(108, 108+∞, 56)-Net over F2 — Constructive and digital
Digital (108, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (108, 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
(108, 108+∞, 65)-Net over F2 — Digital
Digital (108, m, 65)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (108, 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
(108, 108+∞, 118)-Net in Base 2 — Upper bound on s
There is no (108, m, 119)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (108, 823, 119)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2823, 119, S2, 7, 715), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 15326 288425 121365 957052 239839 505675 286050 305728 856950 767708 900817 252405 346614 640116 524624 782882 711279 038368 174855 480318 946063 945014 527549 249847 533467 556044 562540 092891 379500 226603 717862 588359 735724 884691 905269 917594 540960 328605 659981 656850 912449 134592 / 179 > 2823 [i]
- extracting embedded OOA [i] would yield OOA(2823, 119, S2, 7, 715), but