Best Known (178−65, 178, s)-Nets in Base 2
(178−65, 178, 68)-Net over F2 — Constructive and digital
Digital (113, 178, 68)-net over F2, using
- 6 times m-reduction [i] based on digital (113, 184, 68)-net over F2, using
- trace code for nets [i] based on digital (21, 92, 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, 92, 34)-net over F4, using
(178−65, 178, 70)-Net in Base 2 — Constructive
(113, 178, 70)-net in base 2, using
- trace code for nets [i] based on (24, 89, 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
(178−65, 178, 103)-Net over F2 — Digital
Digital (113, 178, 103)-net over F2, using
(178−65, 178, 545)-Net in Base 2 — Upper bound on s
There is no (113, 178, 546)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 177, 546)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 195048 678195 293623 310235 932959 707292 978942 821104 576337 > 2177 [i]