MinT - Best Known (235−47, 235, s)-Nets in Base 2

Best Known (235−47, 235, s)-Nets in Base 2

(235−47, 235, 263)-Net over F₂ — Constructive and digital

Digital (188, 235, 263)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 3)-net over F₂, using
  - net from sequence [i] based on digital (0, 2)-sequence over F₂, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 0 and N(F) ≥ 3, using
      - the rational function field F₂(x) [i]
    - Niederreiter sequence [i]
    - Sobol sequence [i]
2. digital (165, 212, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 53, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(235−47, 235, 593)-Net over F₂ — Digital

Digital (188, 235, 593)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−47, 235, 10858)-Net in Base 2 — Upper bound on s

There is no (188, 235, 10859)-net in base 2, because

1 times m-reduction [i] would yield (188, 234, 10859)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27660 102732 230130 969675 937822 694413 890864 698836 021622 072463 853274 330184 > 2²³⁴ [i]