MinT - Best Known (136, 136+109, s)-Nets in Base 2

Best Known (136, 136+109, s)-Nets in Base 2

(136, 136+109, 66)-Net over F₂ — Constructive and digital

Digital (136, 245, 66)-net over F₂, using

7 times m-reduction [i] based on digital (136, 252, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 97, 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, 155, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(136, 136+109, 86)-Net over F₂ — Digital

Digital (136, 245, 86)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+109, 405)-Net in Base 2 — Upper bound on s

There is no (136, 245, 406)-net in base 2, because

1 times m-reduction [i] would yield (136, 244, 406)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29 175834 410091 517089 229127 752626 438513 917061 251138 217298 353038 281573 267168 > 2²⁴⁴ [i]