MinT - Best Known (230−89, 230, s)-Nets in Base 2

Best Known (230−89, 230, s)-Nets in Base 2

(230−89, 230, 75)-Net over F₂ — Constructive and digital

Digital (141, 230, 75)-net over F₂, using

7 times m-reduction [i] based on digital (141, 237, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 87, 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, 150, 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]

(230−89, 230, 112)-Net over F₂ — Digital

Digital (141, 230, 112)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−89, 230, 573)-Net in Base 2 — Upper bound on s

There is no (141, 230, 574)-net in base 2, because

1 times m-reduction [i] would yield (141, 229, 574)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 880 840940 725303 479449 575689 717323 218028 982010 293393 000424 992940 261273 > 2²²⁹ [i]