MinT - Best Known (90−27, 90, s)-Nets in Base 2

Best Known (90−27, 90, s)-Nets in Base 2

(90−27, 90, 72)-Net over F₂ — Constructive and digital

Digital (63, 90, 72)-net over F₂, using

trace code for nets [i] based on digital (3, 30, 24)-net over F₈, using
- net from sequence [i] based on digital (3, 23)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
    - the Klein quartic over F₈ [i]

(90−27, 90, 96)-Net over F₂ — Digital

Digital (63, 90, 96)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−27, 90, 633)-Net in Base 2 — Upper bound on s

There is no (63, 90, 634)-net in base 2, because

1 times m-reduction [i] would yield (63, 89, 634)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 627 285961 151851 723874 582844 > 2⁸⁹ [i]