MinT - Best Known (69−27, 69, s)-Nets in Base 2

Best Known (69−27, 69, s)-Nets in Base 2

(69−27, 69, 42)-Net over F₂ — Constructive and digital

Digital (42, 69, 42)-net over F₂, using

1 times m-reduction [i] based on digital (42, 70, 42)-net over F₂, using
- trace code for nets [i] based on digital (7, 35, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(69−27, 69, 43)-Net over F₂ — Digital

Digital (42, 69, 43)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−27, 69, 194)-Net in Base 2 — Upper bound on s

There is no (42, 69, 195)-net in base 2, because

1 times m-reduction [i] would yield (42, 68, 195)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 303 266134 762103 637504 > 2⁶⁸ [i]