MinT - Best Known (68, 68+51, s)-Nets in Base 3

Best Known (68, 68+51, s)-Nets in Base 3

(68, 68+51, 80)-Net over F₃ — Constructive and digital

Digital (68, 119, 80)-net over F₃, using

1 times m-reduction [i] based on digital (68, 120, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 60, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(68, 68+51, 100)-Net over F₃ — Digital

Digital (68, 119, 100)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+51, 884)-Net in Base 3 — Upper bound on s

There is no (68, 119, 885)-net in base 3, because

1 times m-reduction [i] would yield (68, 118, 885)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 200 412301 465501 960707 426432 798161 583378 303865 472254 419723 > 3¹¹⁸ [i]