MinT - Best Known (227−87, 227, s)-Nets in Base 2

Best Known (227−87, 227, s)-Nets in Base 2

(227−87, 227, 75)-Net over F₂ — Constructive and digital

Digital (140, 227, 75)-net over F₂, using

7 times m-reduction [i] based on digital (140, 234, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 86, 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, 148, 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]

(227−87, 227, 113)-Net over F₂ — Digital

Digital (140, 227, 113)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−87, 227, 583)-Net in Base 2 — Upper bound on s

There is no (140, 227, 584)-net in base 2, because

1 times m-reduction [i] would yield (140, 226, 584)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 108 884571 738866 631206 146943 921238 242318 749898 568521 072632 294986 625560 > 2²²⁶ [i]