MinT - Best Known (68−22, 68, s)-Nets in Base 2

Best Known (68−22, 68, s)-Nets in Base 2

(68−22, 68, 56)-Net over F₂ — Constructive and digital

Digital (46, 68, 56)-net over F₂, using

trace code for nets [i] based on digital (12, 34, 28)-net over F₄, using
- net from sequence [i] based on digital (12, 27)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(68−22, 68, 66)-Net over F₂ — Digital

Digital (46, 68, 66)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−22, 68, 340)-Net in Base 2 — Upper bound on s

There is no (46, 68, 341)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 297 836971 508081 707008 > 2⁶⁸ [i]