MinT - Best Known (67, 67+31, s)-Nets in Base 2

Best Known (67, 67+31, s)-Nets in Base 2

(67, 67+31, 66)-Net over F₂ — Constructive and digital

Digital (67, 98, 66)-net over F₂, using

6 times m-reduction [i] based on digital (67, 104, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 52, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(67, 67+31, 89)-Net over F₂ — Digital

Digital (67, 98, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+31, 546)-Net in Base 2 — Upper bound on s

There is no (67, 98, 547)-net in base 2, because

1 times m-reduction [i] would yield (67, 97, 547)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 160642 375489 971415 145231 455856 > 2⁹⁷ [i]