MinT - Best Known (110−33, 110, s)-Nets in Base 2

Best Known (110−33, 110, s)-Nets in Base 2

(110−33, 110, 72)-Net over F₂ — Constructive and digital

Digital (77, 110, 72)-net over F₂, using

1 times m-reduction [i] based on digital (77, 111, 72)-net over F₂, using
- trace code for nets [i] based on digital (3, 37, 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]

(110−33, 110, 110)-Net over F₂ — Digital

Digital (77, 110, 110)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−33, 110, 741)-Net in Base 2 — Upper bound on s

There is no (77, 110, 742)-net in base 2, because

1 times m-reduction [i] would yield (77, 109, 742)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 660 055367 182100 617991 406971 901714 > 2¹⁰⁹ [i]