Best Known (89, ∞, s)-Nets in Base 2
(89, ∞, 52)-Net over F2 — Constructive and digital
Digital (89, m, 52)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (89, 51)-sequence over F2, using
- t-expansion [i] based on digital (85, 51)-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 3 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 (85, 51)-sequence over F2, using
(89, ∞, 57)-Net over F2 — Digital
Digital (89, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (89, 56)-sequence over F2, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
(89, ∞, 98)-Net in Base 2 — Upper bound on s
There is no (89, m, 99)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (89, 684, 99)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2684, 99, S2, 7, 595), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 12601 338753 404154 062451 712213 320620 735037 282547 656834 349689 180019 863753 482431 666398 191004 122857 382001 383000 581518 859560 866455 854237 043109 230210 371688 725971 884810 945379 137661 772401 294097 714823 404107 559822 426112 / 149 > 2684 [i]
- extracting embedded OOA [i] would yield OOA(2684, 99, S2, 7, 595), but