MinT - Best Known (234−97, 234, s)-Nets in Base 2

Best Known (234−97, 234, s)-Nets in Base 2

(234−97, 234, 69)-Net over F₂ — Constructive and digital

Digital (137, 234, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 67, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 167, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(234−97, 234, 97)-Net over F₂ — Digital

Digital (137, 234, 97)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−97, 234, 474)-Net in Base 2 — Upper bound on s

There is no (137, 234, 475)-net in base 2, because

1 times m-reduction [i] would yield (137, 233, 475)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 14333 255190 175960 845072 725572 523877 270084 593310 024523 699638 121617 166250 > 2²³³ [i]