Best Known (50, ∞, s)-Nets in Base 2
(50, ∞, 35)-Net over F2 — Constructive and digital
Digital (50, m, 35)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (50, 34)-sequence over F2, using
- t-expansion [i] based on digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (48, 34)-sequence over F2, using
(50, ∞, 40)-Net over F2 — Digital
Digital (50, m, 40)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
(50, ∞, 58)-Net in Base 2 — Upper bound on s
There is no (50, m, 59)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (50, 405, 59)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2405, 59, S2, 7, 355), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 8139 251615 729935 874595 457230 153493 427887 232658 997712 512442 238291 816121 304597 083445 472642 153538 994444 685966 798058 100099 121152 / 89 > 2405 [i]
- extracting embedded OOA [i] would yield OOA(2405, 59, S2, 7, 355), but