Best Known (110, ∞, s)-Nets in Base 2
(110, ∞, 57)-Net over F2 — Constructive and digital
Digital (110, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (110, 56)-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 8 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]
(110, ∞, 72)-Net over F2 — Digital
Digital (110, m, 72)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (110, 71)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 110 and N(F) ≥ 72, using
(110, ∞, 120)-Net in Base 2 — Upper bound on s
There is no (110, m, 121)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (110, 837, 121)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2837, 121, S2, 7, 727), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 123 720064 927811 832403 089146 593475 302279 917913 223777 218927 187062 713983 796503 124545 968371 628028 362394 581854 833079 826078 790721 394438 470556 987345 922512 499558 983960 615617 010470 196281 792581 449589 188313 378572 835561 139469 096019 991581 106361 218349 960299 968955 678720 / 91 > 2837 [i]
- extracting embedded OOA [i] would yield OOA(2837, 121, S2, 7, 727), but