Best Known (128−40, 128, s)-Nets in Base 2
(128−40, 128, 68)-Net over F2 — Constructive and digital
Digital (88, 128, 68)-net over F2, using
- 6 times m-reduction [i] based on digital (88, 134, 68)-net over F2, using
- trace code for nets [i] based on digital (21, 67, 34)-net over F4, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- T5 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- trace code for nets [i] based on digital (21, 67, 34)-net over F4, using
(128−40, 128, 70)-Net in Base 2 — Constructive
(88, 128, 70)-net in base 2, using
- trace code for nets [i] based on (24, 64, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
- base expansion [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]
- base expansion [i] based on digital (48, 34)-sequence over F2, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
(128−40, 128, 113)-Net over F2 — Digital
Digital (88, 128, 113)-net over F2, using
(128−40, 128, 672)-Net in Base 2 — Upper bound on s
There is no (88, 128, 673)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 346 719284 056149 367034 485213 412223 156616 > 2128 [i]