Best Known (70, ∞, s)-Nets in Base 2
(70, ∞, 49)-Net over F2 — Constructive and digital
Digital (70, m, 49)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(70, ∞, 79)-Net in Base 2 — Upper bound on s
There is no (70, m, 80)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (70, 472, 80)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2472, 80, S2, 6, 402), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5 463059 963052 986404 917795 152138 041035 060046 749663 699514 003524 679344 400542 934589 835210 535680 361110 949529 697129 462461 590067 033957 438754 499286 007808 / 403 > 2472 [i]
- extracting embedded OOA [i] would yield OOA(2472, 80, S2, 6, 402), but