MinT - Best Known (222−85, 222, s)-Nets in Base 2

Best Known (222−85, 222, s)-Nets in Base 2

(222−85, 222, 75)-Net over F₂ — Constructive and digital

Digital (137, 222, 75)-net over F₂, using

3 times m-reduction [i] based on digital (137, 225, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 83, 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, 142, 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]

(222−85, 222, 111)-Net over F₂ — Digital

Digital (137, 222, 111)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−85, 222, 573)-Net in Base 2 — Upper bound on s

There is no (137, 222, 574)-net in base 2, because

1 times m-reduction [i] would yield (137, 221, 574)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 427546 997603 864267 928403 353719 180917 788130 507314 192426 265396 200526 > 2²²¹ [i]