MinT - Best Known (210−48, 210, s)-Nets in Base 2

Best Known (210−48, 210, s)-Nets in Base 2

(210−48, 210, 198)-Net over F₂ — Constructive and digital

Digital (162, 210, 198)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 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 (138, 186, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 62, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(210−48, 210, 369)-Net over F₂ — Digital

Digital (162, 210, 369)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−48, 210, 4185)-Net in Base 2 — Upper bound on s

There is no (162, 210, 4186)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1652 578758 281699 038577 588911 058987 970426 201103 041930 114897 802669 > 2²¹⁰ [i]