MinT - Best Known (221−85, 221, s)-Nets in Base 2

Best Known (221−85, 221, s)-Nets in Base 2

(221−85, 221, 75)-Net over F₂ — Constructive and digital

Digital (136, 221, 75)-net over F₂, using

1 times m-reduction [i] based on digital (136, 222, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 82, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 140, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(221−85, 221, 110)-Net over F₂ — Digital

Digital (136, 221, 110)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−85, 221, 563)-Net in Base 2 — Upper bound on s

There is no (136, 221, 564)-net in base 2, because

1 times m-reduction [i] would yield (136, 220, 564)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 752725 072959 154677 400095 040474 809902 956713 000599 200888 250793 677278 > 2²²⁰ [i]