MinT - Best Known (180−47, 180, s)-Nets in Base 2

Best Known (180−47, 180, s)-Nets in Base 2

(180−47, 180, 144)-Net over F₂ — Constructive and digital

Digital (133, 180, 144)-net over F₂, using

3 times m-reduction [i] based on digital (133, 183, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 61, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]

(180−47, 180, 229)-Net over F₂ — Digital

Digital (133, 180, 229)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−47, 180, 2042)-Net in Base 2 — Upper bound on s

There is no (133, 180, 2043)-net in base 2, because

1 times m-reduction [i] would yield (133, 179, 2043)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 771496 327468 281501 956853 517238 381733 831015 769148 066756 > 2¹⁷⁹ [i]