MinT - Best Known (67−27, 67, s)-Nets in Base 2

Best Known (67−27, 67, s)-Nets in Base 2

(67−27, 67, 34)-Net over F₂ — Constructive and digital

Digital (40, 67, 34)-net over F₂, using

3 times m-reduction [i] based on digital (40, 70, 34)-net over F₂, using
- trace code for nets [i] based on digital (5, 35, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]

(67−27, 67, 40)-Net over F₂ — Digital

Digital (40, 67, 40)-net over F₂, using

1 times m-reduction [i] based on digital (40, 68, 40)-net over F₂, using
- trace code for nets [i] based on digital (6, 34, 20)-net over F₄, using
  - net from sequence [i] based on digital (6, 19)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 6 and N(F) ≥ 20, using
      - Drinfeld modules of rank 1 [i]

(67−27, 67, 173)-Net in Base 2 — Upper bound on s

There is no (40, 67, 174)-net in base 2, because

1 times m-reduction [i] would yield (40, 66, 174)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 78 406938 414606 439263 > 2⁶⁶ [i]