MinT - Best Known (189−75, 189, s)-Nets in Base 2

Best Known (189−75, 189, s)-Nets in Base 2

(189−75, 189, 66)-Net over F₂ — Constructive and digital

Digital (114, 189, 66)-net over F₂, using

9 times m-reduction [i] based on digital (114, 198, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 99, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(189−75, 189, 90)-Net over F₂ — Digital

Digital (114, 189, 90)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−75, 189, 443)-Net in Base 2 — Upper bound on s

There is no (114, 189, 444)-net in base 2, because

1 times m-reduction [i] would yield (114, 188, 444)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 403 769736 941258 605606 460278 319239 520347 358039 904396 759474 > 2¹⁸⁸ [i]