MinT - Best Known (68, 68+48, s)-Nets in Base 3

Best Known (68, 68+48, s)-Nets in Base 3

(68, 68+48, 80)-Net over F₃ — Constructive and digital

Digital (68, 116, 80)-net over F₃, using

4 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+48, 108)-Net over F₃ — Digital

Digital (68, 116, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 58, 54)-net over F₉, using
- net from sequence [i] based on digital (10, 53)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 10 and N(F) ≥ 54, using
    - a curve from the manYPoints database [i]

(68, 68+48, 968)-Net in Base 3 — Upper bound on s

There is no (68, 116, 969)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 445723 404786 541243 217238 204001 032745 441226 665766 346209 > 3¹¹⁶ [i]