MinT - Best Known (218−91, 218, s)-Nets in Base 2

Best Known (218−91, 218, s)-Nets in Base 2

(218−91, 218, 66)-Net over F₂ — Constructive and digital

Digital (127, 218, 66)-net over F₂, using

6 times m-reduction [i] based on digital (127, 224, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 112, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(218−91, 218, 90)-Net over F₂ — Digital

Digital (127, 218, 90)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−91, 218, 435)-Net in Base 2 — Upper bound on s

There is no (127, 218, 436)-net in base 2, because

1 times m-reduction [i] would yield (127, 217, 436)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 220391 429544 012659 396673 210716 216197 105941 607481 088081 724193 772876 > 2²¹⁷ [i]