MinT - Best Known (139, 139+91, s)-Nets in Base 2

Best Known (139, 139+91, s)-Nets in Base 2

(139, 139+91, 75)-Net over F₂ — Constructive and digital

Digital (139, 230, 75)-net over F₂, using

1 times m-reduction [i] based on digital (139, 231, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 85, 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, 146, 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]

(139, 139+91, 107)-Net over F₂ — Digital

Digital (139, 230, 107)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+91, 536)-Net in Base 2 — Upper bound on s

There is no (139, 230, 537)-net in base 2, because

1 times m-reduction [i] would yield (139, 229, 537)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 920 982199 673853 627249 258638 896500 924263 832215 520352 762106 632145 361792 > 2²²⁹ [i]