MinT - Best Known (137, 137+51, s)-Nets in Base 2

Best Known (137, 137+51, s)-Nets in Base 2

(137, 137+51, 144)-Net over F₂ — Constructive and digital

Digital (137, 188, 144)-net over F₂, using

1 times m-reduction [i] based on digital (137, 189, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 63, 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]

(137, 137+51, 216)-Net over F₂ — Digital

Digital (137, 188, 216)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+51, 1780)-Net in Base 2 — Upper bound on s

There is no (137, 188, 1781)-net in base 2, because

1 times m-reduction [i] would yield (137, 187, 1781)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 198 229713 912809 269941 617583 834065 127717 055993 221228 573288 > 2¹⁸⁷ [i]