MinT - Best Known (109−32, 109, s)-Nets in Base 2

Best Known (109−32, 109, s)-Nets in Base 2

(109−32, 109, 75)-Net over F₂ — Constructive and digital

Digital (77, 109, 75)-net over F₂, using

2¹ times duplication [i] based on digital (76, 108, 75)-net over F₂, using
- trace code for nets [i] based on digital (4, 36, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]

(109−32, 109, 116)-Net over F₂ — Digital

Digital (77, 109, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−32, 109, 741)-Net in Base 2 — Upper bound on s

There is no (77, 109, 742)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 660 055367 182100 617991 406971 901714 > 2¹⁰⁹ [i]