MinT - Best Known (40, 68, s)-Nets in Base 2

Best Known (40, 68, s)-Nets in Base 2

(40, 68, 34)-Net over F₂ — Constructive and digital

Digital (40, 68, 34)-net over F₂, using

2 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]

(40, 68, 40)-Net over F₂ — Digital

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]

(40, 68, 155)-Net in Base 2 — Upper bound on s

There is no (40, 68, 156)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 301 163591 483081 957708 > 2⁶⁸ [i]