MinT - Best Known (231−101, 231, s)-Nets in Base 2

Best Known (231−101, 231, s)-Nets in Base 2

(231−101, 231, 66)-Net over F₂ — Constructive and digital

Digital (130, 231, 66)-net over F₂, using

3 times m-reduction [i] based on digital (130, 234, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 91, 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 (39, 143, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(231−101, 231, 85)-Net over F₂ — Digital

Digital (130, 231, 85)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−101, 231, 402)-Net in Base 2 — Upper bound on s

There is no (130, 231, 403)-net in base 2, because

1 times m-reduction [i] would yield (130, 230, 403)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1734 994748 689103 027662 946171 895644 702845 405360 618205 538760 842700 578268 > 2²³⁰ [i]