MinT - Best Known (252−105, 252, s)-Nets in Base 2

Best Known (252−105, 252, s)-Nets in Base 2

(252−105, 252, 75)-Net over F₂ — Constructive and digital

Digital (147, 252, 75)-net over F₂, using

3 times m-reduction [i] based on digital (147, 255, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 93, 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, 162, 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]

(252−105, 252, 102)-Net over F₂ — Digital

Digital (147, 252, 102)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−105, 252, 501)-Net in Base 2 — Upper bound on s

There is no (147, 252, 502)-net in base 2, because

1 times m-reduction [i] would yield (147, 251, 502)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3961 480351 539026 289623 838098 817876 164917 477524 477775 512413 519049 266395 601027 > 2²⁵¹ [i]