MinT - Best Known (220−51, 220, s)-Nets in Base 2

Best Known (220−51, 220, s)-Nets in Base 2

(220−51, 220, 198)-Net over F₂ — Constructive and digital

Digital (169, 220, 198)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 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 (144, 195, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 65, 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]

(220−51, 220, 370)-Net over F₂ — Digital

Digital (169, 220, 370)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−51, 220, 4375)-Net in Base 2 — Upper bound on s

There is no (169, 220, 4376)-net in base 2, because

1 times m-reduction [i] would yield (169, 219, 4376)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 845970 493174 164602 698320 148225 303617 304286 363901 198763 845004 265232 > 2²¹⁹ [i]