MinT - Best Known (68−26, 68, s)-Nets in Base 2

Best Known (68−26, 68, s)-Nets in Base 2

(68−26, 68, 42)-Net over F₂ — Constructive and digital

Digital (42, 68, 42)-net over F₂, using

2 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]

(68−26, 68, 45)-Net over F₂ — Digital

Digital (42, 68, 45)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−26, 68, 194)-Net in Base 2 — Upper bound on s

There is no (42, 68, 195)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 303 266134 762103 637504 > 2⁶⁸ [i]