Best Known (51, 51+85, s)-Nets in Base 2
(51, 51+85, 36)-Net over F2 — Constructive and digital
Digital (51, 136, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-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 3 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]
(51, 51+85, 40)-Net over F2 — Digital
Digital (51, 136, 40)-net over F2, using
- t-expansion [i] based on digital (50, 136, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(51, 51+85, 100)-Net in Base 2 — Upper bound on s
There is no (51, 136, 101)-net in base 2, because
- 1 times m-reduction [i] would yield (51, 135, 101)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52454 092216 849927 143205 531096 225212 135632 > 2135 [i]