Best Known (219−79, 219, s)-Nets in Base 2
(219−79, 219, 76)-Net over F2 — Constructive and digital
Digital (140, 219, 76)-net over F2, using
- 3 times m-reduction [i] based on digital (140, 222, 76)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (45, 86, 34)-net over F2, using
- net from sequence [i] based on digital (45, 33)-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 1 place 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]
- net from sequence [i] based on digital (45, 33)-sequence over F2, using
- digital (54, 136, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- digital (45, 86, 34)-net over F2, using
- (u, u+v)-construction [i] based on
(219−79, 219, 86)-Net in Base 2 — Constructive
(140, 219, 86)-net in base 2, using
- 1 times m-reduction [i] based on (140, 220, 86)-net in base 2, using
- trace code for nets [i] based on (30, 110, 43)-net in base 4, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- t-expansion [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]
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
- trace code for nets [i] based on (30, 110, 43)-net in base 4, using
(219−79, 219, 126)-Net over F2 — Digital
Digital (140, 219, 126)-net over F2, using
(219−79, 219, 685)-Net in Base 2 — Upper bound on s
There is no (140, 219, 686)-net in base 2, because
- 1 times m-reduction [i] would yield (140, 218, 686)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 432900 567581 338837 815912 060908 460863 051575 791059 260011 905945 203392 > 2218 [i]