Best Known (84, ∞, s)-Nets in Base 2
(84, ∞, 51)-Net over F2 — Constructive and digital
Digital (84, m, 51)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (84, 50)-sequence over F2, using
- t-expansion [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(84, ∞, 57)-Net over F2 — Digital
Digital (84, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (84, 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
(84, ∞, 93)-Net in Base 2 — Upper bound on s
There is no (84, m, 94)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (84, 649, 94)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2649, 94, S2, 7, 565), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 756854 149176 110720 429946 675326 409927 303887 890610 449109 763921 088409 274958 937913 074292 692864 443816 106766 479415 450544 429039 955527 848819 528658 654137 481006 078667 260746 892540 275632 039347 834003 404607 193088 / 283 > 2649 [i]
- extracting embedded OOA [i] would yield OOA(2649, 94, S2, 7, 565), but