Best Known (138, 138+75, s)-Nets in Base 2
(138, 138+75, 76)-Net over F2 — Constructive and digital
Digital (138, 213, 76)-net over F2, using
- 3 times m-reduction [i] based on digital (138, 216, 76)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (45, 84, 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, 132, 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, 84, 34)-net over F2, using
- (u, u+v)-construction [i] based on
(138, 138+75, 86)-Net in Base 2 — Constructive
(138, 213, 86)-net in base 2, using
- 3 times m-reduction [i] based on (138, 216, 86)-net in base 2, using
- trace code for nets [i] based on (30, 108, 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, 108, 43)-net in base 4, using
(138, 138+75, 130)-Net over F2 — Digital
Digital (138, 213, 130)-net over F2, using
(138, 138+75, 724)-Net in Base 2 — Upper bound on s
There is no (138, 213, 725)-net in base 2, because
- 1 times m-reduction [i] would yield (138, 212, 725)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6814 564745 250309 895983 507073 949925 592691 413481 226231 587378 708460 > 2212 [i]