Best Known (210−127, 210, s)-Nets in Base 2
(210−127, 210, 51)-Net over F2 — Constructive and digital
Digital (83, 210, 51)-net over F2, using
- t-expansion [i] based on digital (80, 210, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
(210−127, 210, 57)-Net over F2 — Digital
Digital (83, 210, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
(210−127, 210, 162)-Net in Base 2 — Upper bound on s
There is no (83, 210, 163)-net in base 2, because
- 1 times m-reduction [i] would yield (83, 209, 163)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1039 434005 956716 656136 857109 850742 857458 608189 578078 938210 592832 > 2209 [i]