Best Known (215−77, 215, s)-Nets in Base 2
(215−77, 215, 76)-Net over F2 — Constructive and digital
Digital (138, 215, 76)-net over F2, using
- 1 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
(215−77, 215, 86)-Net in Base 2 — Constructive
(138, 215, 86)-net in base 2, using
- 1 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
(215−77, 215, 126)-Net over F2 — Digital
Digital (138, 215, 126)-net over F2, using
(215−77, 215, 690)-Net in Base 2 — Upper bound on s
There is no (138, 215, 691)-net in base 2, because
- 1 times m-reduction [i] would yield (138, 214, 691)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 27359 482331 679895 748046 925570 752599 003637 110126 369996 677432 309068 > 2214 [i]