MinT - Best Known (140, 140+89, s)-Nets in Base 2

Best Known (140, 140+89, s)-Nets in Base 2

(140, 140+89, 75)-Net over F₂ — Constructive and digital

Digital (140, 229, 75)-net over F₂, using

5 times m-reduction [i] based on digital (140, 234, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 86, 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, 148, 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]

(140, 140+89, 110)-Net over F₂ — Digital

Digital (140, 229, 110)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+89, 563)-Net in Base 2 — Upper bound on s

There is no (140, 229, 564)-net in base 2, because

1 times m-reduction [i] would yield (140, 228, 564)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 437 506380 082242 437565 698944 450917 211273 138936 783724 525653 135382 812604 > 2²²⁸ [i]