Best Known (210−81, 210, s)-Nets in Base 2
(210−81, 210, 69)-Net over F2 — Constructive and digital
Digital (129, 210, 69)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 59, 20)-net over F2, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 19 and N(F) ≥ 20, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- digital (70, 151, 49)-net over F2, 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]
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- digital (19, 59, 20)-net over F2, using
(210−81, 210, 70)-Net in Base 2 — Constructive
(129, 210, 70)-net in base 2, using
- trace code for nets [i] based on (24, 105, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
- base 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]
- base expansion [i] based on digital (48, 34)-sequence over F2, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
(210−81, 210, 104)-Net over F2 — Digital
Digital (129, 210, 104)-net over F2, using
(210−81, 210, 532)-Net in Base 2 — Upper bound on s
There is no (129, 210, 533)-net in base 2, because
- 1 times m-reduction [i] would yield (129, 209, 533)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 830 630970 093486 567175 616963 474709 138419 395984 355298 356703 053280 > 2209 [i]