MinT - Best Known (75−45, 75, s)-Nets in Base 2

Best Known (75−45, 75, s)-Nets in Base 2

(75−45, 75, 21)-Net over F₂ — Constructive and digital

Digital (30, 75, 21)-net over F₂, using

t-expansion [i] based on digital (21, 75, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(75−45, 75, 25)-Net over F₂ — Digital

Digital (30, 75, 25)-net over F₂, using

t-expansion [i] based on digital (28, 75, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(75−45, 75, 65)-Net in Base 2 — Upper bound on s

There is no (30, 75, 66)-net in base 2, because

1 times m-reduction [i] would yield (30, 74, 66)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23729 704912 474978 222048 > 2⁷⁴ [i]