MinT - Best Known (237−101, 237, s)-Nets in Base 2

Best Known (237−101, 237, s)-Nets in Base 2

(237−101, 237, 67)-Net over F₂ — Constructive and digital

Digital (136, 237, 67)-net over F₂, using

3 times m-reduction [i] based on digital (136, 240, 67)-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 (45, 149, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(237−101, 237, 92)-Net over F₂ — Digital

Digital (136, 237, 92)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−101, 237, 443)-Net in Base 2 — Upper bound on s

There is no (136, 237, 444)-net in base 2, because

1 times m-reduction [i] would yield (136, 236, 444)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 115250 881254 843134 123060 215584 287420 383132 863707 279546 921554 145847 456633 > 2²³⁶ [i]