Best Known (52, ∞, s)-Nets in Base 2
(52, ∞, 36)-Net over F2 — Constructive and digital
Digital (52, m, 36)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (52, 35)-sequence over F2, using
- t-expansion [i] based on digital (51, 35)-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 3 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 (51, 35)-sequence over F2, using
(52, ∞, 40)-Net over F2 — Digital
Digital (52, m, 40)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (52, 39)-sequence over F2, using
- t-expansion [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
- t-expansion [i] based on digital (50, 39)-sequence over F2, using
(52, ∞, 60)-Net in Base 2 — Upper bound on s
There is no (52, m, 61)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (52, 419, 61)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2419, 61, S2, 7, 367), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 32 999913 962009 210059 679612 176995 930308 233961 773576 918466 689667 860098 337098 516662 383593 445385 150587 019591 753304 289166 046548 459520 / 23 > 2419 [i]
- extracting embedded OOA [i] would yield OOA(2419, 61, S2, 7, 367), but