MinT - Best Known (227−99, 227, s)-Nets in Base 2

Best Known (227−99, 227, s)-Nets in Base 2

(227−99, 227, 66)-Net over F₂ — Constructive and digital

Digital (128, 227, 66)-net over F₂, using

1 times m-reduction [i] based on digital (128, 228, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 89, 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 (39, 139, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(227−99, 227, 84)-Net over F₂ — Digital

Digital (128, 227, 84)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−99, 227, 399)-Net in Base 2 — Upper bound on s

There is no (128, 227, 400)-net in base 2, because

1 times m-reduction [i] would yield (128, 226, 400)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 116 677308 112093 916509 891673 383365 577823 754875 630216 488666 133156 628552 > 2²²⁶ [i]