Best Known (212−129, 212, s)-Nets in Base 2
(212−129, 212, 51)-Net over F2 — Constructive and digital
Digital (83, 212, 51)-net over F2, using
- t-expansion [i] based on digital (80, 212, 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
(212−129, 212, 57)-Net over F2 — Digital
Digital (83, 212, 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
(212−129, 212, 161)-Net in Base 2 — Upper bound on s
There is no (83, 212, 162)-net in base 2, because
- 1 times m-reduction [i] would yield (83, 211, 162)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4050 427566 614345 838109 877885 358827 757874 525378 721761 134177 122962 > 2211 [i]