MinT - Best Known (212−57, 212, s)-Nets in Base 2

Best Known (212−57, 212, s)-Nets in Base 2

(212−57, 212, 144)-Net over F₂ — Constructive and digital

Digital (155, 212, 144)-net over F₂, using

4 times m-reduction [i] based on digital (155, 216, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 72, 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]

(212−57, 212, 244)-Net over F₂ — Digital

Digital (155, 212, 244)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−57, 212, 2055)-Net in Base 2 — Upper bound on s

There is no (155, 212, 2056)-net in base 2, because

1 times m-reduction [i] would yield (155, 211, 2056)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3321 692248 708942 242922 605253 854941 101338 136987 961103 860425 373952 > 2²¹¹ [i]