MinT - Best Known (72−30, 72, s)-Nets in Base 2

Best Known (72−30, 72, s)-Nets in Base 2

(72−30, 72, 34)-Net over F₂ — Constructive and digital

Digital (42, 72, 34)-net over F₂, using

2 times m-reduction [i] based on digital (42, 74, 34)-net over F₂, using
- trace code for nets [i] based on digital (5, 37, 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]

(72−30, 72, 40)-Net over F₂ — Digital

Digital (42, 72, 40)-net over F₂, using

trace code for nets [i] based on digital (6, 36, 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]

(72−30, 72, 158)-Net in Base 2 — Upper bound on s

There is no (42, 72, 159)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5097 877774 825083 175850 > 2⁷² [i]