Best Known (49, ∞, s)-Nets in Base 2
(49, ∞, 35)-Net over F2 — Constructive and digital
Digital (49, m, 35)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (49, 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
(49, ∞, 36)-Net over F2 — Digital
Digital (49, m, 36)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (49, 35)-sequence over F2, using
- t-expansion [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
- t-expansion [i] based on digital (47, 35)-sequence over F2, using
(49, ∞, 57)-Net in Base 2 — Upper bound on s
There is no (49, m, 58)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (49, 398, 58)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2398, 58, S2, 7, 349), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 127 821368 965301 975187 967999 014149 087779 320436 745046 563885 117638 148762 057289 833639 134167 444987 366822 656077 207012 651000 922112 / 175 > 2398 [i]
- extracting embedded OOA [i] would yield OOA(2398, 58, S2, 7, 349), but