MinT - Best Known (68, 68+57, s)-Nets in Base 3

Best Known (68, 68+57, s)-Nets in Base 3

(68, 68+57, 64)-Net over F₃ — Constructive and digital

Digital (68, 125, 64)-net over F₃, using

1 times m-reduction [i] based on digital (68, 126, 64)-net over F₃, using
- trace code for nets [i] based on digital (5, 63, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+57, 87)-Net over F₃ — Digital

Digital (68, 125, 87)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+57, 705)-Net in Base 3 — Upper bound on s

There is no (68, 125, 706)-net in base 3, because

1 times m-reduction [i] would yield (68, 124, 706)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 147535 856626 552940 990569 635911 759387 670942 613700 485924 379673 > 3¹²⁴ [i]