MinT - Best Known (255−109, 255, s)-Nets in Base 2

Best Known (255−109, 255, s)-Nets in Base 2

(255−109, 255, 70)-Net over F₂ — Constructive and digital

Digital (146, 255, 70)-net over F₂, using

1 times m-reduction [i] based on digital (146, 256, 70)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (70, 180, 49)-net over F₂, using
    - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(255−109, 255, 98)-Net over F₂ — Digital

Digital (146, 255, 98)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−109, 255, 470)-Net in Base 2 — Upper bound on s

There is no (146, 255, 471)-net in base 2, because

1 times m-reduction [i] would yield (146, 254, 471)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29204 624772 146489 935573 989907 894869 057136 281872 765569 270908 345644 379638 002774 > 2²⁵⁴ [i]