MinT - Best Known (70, 70+31, s)-Nets in Base 2

Best Known (70, 70+31, s)-Nets in Base 2

(70, 70+31, 66)-Net over F₂ — Constructive and digital

Digital (70, 101, 66)-net over F₂, using

9 times m-reduction [i] based on digital (70, 110, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 55, 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]

(70, 70+31, 98)-Net over F₂ — Digital

Digital (70, 101, 98)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+31, 630)-Net in Base 2 — Upper bound on s

There is no (70, 101, 631)-net in base 2, because

1 times m-reduction [i] would yield (70, 100, 631)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 270301 285195 463121 657376 143140 > 2¹⁰⁰ [i]