Best Known (222−79, 222, s)-Nets in Base 2
(222−79, 222, 77)-Net over F2 — Constructive and digital
Digital (143, 222, 77)-net over F2, using
- 3 times m-reduction [i] based on digital (143, 225, 77)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (48, 89, 35)-net over F2, using
- net from sequence [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]
- net from sequence [i] based on digital (48, 34)-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 (48, 89, 35)-net over F2, using
- (u, u+v)-construction [i] based on
(222−79, 222, 86)-Net in Base 2 — Constructive
(143, 222, 86)-net in base 2, using
- 4 times m-reduction [i] based on (143, 226, 86)-net in base 2, using
- trace code for nets [i] based on (30, 113, 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, 113, 43)-net in base 4, using
(222−79, 222, 131)-Net over F2 — Digital
Digital (143, 222, 131)-net over F2, using
(222−79, 222, 725)-Net in Base 2 — Upper bound on s
There is no (143, 222, 726)-net in base 2, because
- 1 times m-reduction [i] would yield (143, 221, 726)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 371376 947244 844972 061845 114396 705080 255227 519061 593270 166226 295544 > 2221 [i]