MinT - Best Known (109−33, 109, s)-Nets in Base 2

Best Known (109−33, 109, s)-Nets in Base 2

(109−33, 109, 72)-Net over F₂ — Constructive and digital

Digital (76, 109, 72)-net over F₂, using

2¹ times duplication [i] based on digital (75, 108, 72)-net over F₂, using
- trace code for nets [i] based on digital (3, 36, 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]

(109−33, 109, 107)-Net over F₂ — Digital

Digital (76, 109, 107)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−33, 109, 708)-Net in Base 2 — Upper bound on s

There is no (76, 109, 709)-net in base 2, because

1 times m-reduction [i] would yield (76, 108, 709)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 325 920915 637290 281866 893212 387620 > 2¹⁰⁸ [i]