MinT - Best Known (187−51, 187, s)-Nets in Base 2

Best Known (187−51, 187, s)-Nets in Base 2

(187−51, 187, 144)-Net over F₂ — Constructive and digital

Digital (136, 187, 144)-net over F₂, using

2¹ times duplication [i] based on digital (135, 186, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 62, 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]

(187−51, 187, 212)-Net over F₂ — Digital

Digital (136, 187, 212)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−51, 187, 1730)-Net in Base 2 — Upper bound on s

There is no (136, 187, 1731)-net in base 2, because

1 times m-reduction [i] would yield (136, 186, 1731)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 98 687555 836563 743858 667966 909385 102629 661416 804705 705376 > 2¹⁸⁶ [i]