MinT - Best Known (218−83, 218, s)-Nets in Base 2

Best Known (218−83, 218, s)-Nets in Base 2

(218−83, 218, 75)-Net over F₂ — Constructive and digital

Digital (135, 218, 75)-net over F₂, using

1 times m-reduction [i] based on digital (135, 219, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 81, 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, 138, 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]

(218−83, 218, 111)-Net over F₂ — Digital

Digital (135, 218, 111)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−83, 218, 574)-Net in Base 2 — Upper bound on s

There is no (135, 218, 575)-net in base 2, because

1 times m-reduction [i] would yield (135, 217, 575)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 221645 187092 940038 790627 856373 129415 687592 798303 323105 132371 129864 > 2²¹⁷ [i]