MinT - Best Known (68, 68+20, s)-Nets in Base 2

Best Known (68, 68+20, s)-Nets in Base 2

(68, 68+20, 135)-Net over F₂ — Constructive and digital

Digital (68, 88, 135)-net over F₂, using

2¹ times duplication [i] based on digital (67, 87, 135)-net over F₂, using
- trace code for nets [i] based on digital (9, 29, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(68, 68+20, 194)-Net over F₂ — Digital

Digital (68, 88, 194)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+20, 2004)-Net in Base 2 — Upper bound on s

There is no (68, 88, 2005)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 310 867760 980955 298193 859404 > 2⁸⁸ [i]