Best Known (68, 68+65, s)-Nets in Base 2
(68, 68+65, 43)-Net over F2 — Constructive and digital
Digital (68, 133, 43)-net over F2, using
- t-expansion [i] based on digital (59, 133, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(68, 68+65, 49)-Net over F2 — Digital
Digital (68, 133, 49)-net over F2, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 68 and N(F) ≥ 49, using
(68, 68+65, 149)-Net in Base 2 — Upper bound on s
There is no (68, 133, 150)-net in base 2, because
- 1 times m-reduction [i] would yield (68, 132, 150)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2132, 150, S2, 64), but
- the linear programming bound shows that M ≥ 7654 730789 395636 393332 135297 086550 086927 777792 / 1 285141 > 2132 [i]
- extracting embedded orthogonal array [i] would yield OA(2132, 150, S2, 64), but