Best Known (109, 109+45, s)-Nets in Base 2
(109, 109+45, 84)-Net over F2 — Constructive and digital
Digital (109, 154, 84)-net over F2, using
- 2 times m-reduction [i] based on digital (109, 156, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 52, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- trace code for nets [i] based on digital (5, 52, 28)-net over F8, using
(109, 109+45, 86)-Net in Base 2 — Constructive
(109, 154, 86)-net in base 2, using
- 4 times m-reduction [i] based on (109, 158, 86)-net in base 2, using
- trace code for nets [i] based on (30, 79, 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, 79, 43)-net in base 4, using
(109, 109+45, 153)-Net over F2 — Digital
Digital (109, 154, 153)-net over F2, using
(109, 109+45, 1090)-Net in Base 2 — Upper bound on s
There is no (109, 154, 1091)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 153, 1091)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11435 370617 023737 652887 027700 967257 565061 043728 > 2153 [i]