MinT - Best Known (157, 157+59, s)-Nets in Base 2

Best Known (157, 157+59, s)-Nets in Base 2

(157, 157+59, 144)-Net over F₂ — Constructive and digital

Digital (157, 216, 144)-net over F₂, using

3 times m-reduction [i] based on digital (157, 219, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 73, 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]

(157, 157+59, 238)-Net over F₂ — Digital

Digital (157, 216, 238)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+59, 1947)-Net in Base 2 — Upper bound on s

There is no (157, 216, 1948)-net in base 2, because

1 times m-reduction [i] would yield (157, 215, 1948)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 52950 500731 816997 144140 142501 682645 576531 029742 384772 535124 416696 > 2²¹⁵ [i]