MinT - Best Known (69, 69+49, s)-Nets in Base 3

Best Known (69, 69+49, s)-Nets in Base 3

(69, 69+49, 80)-Net over F₃ — Constructive and digital

Digital (69, 118, 80)-net over F₃, using

4 times m-reduction [i] based on digital (69, 122, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 61, 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]

(69, 69+49, 108)-Net over F₃ — Digital

Digital (69, 118, 108)-net over F₃, using

trace code for nets [i] based on digital (10, 59, 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]

(69, 69+49, 1014)-Net in Base 3 — Upper bound on s

There is no (69, 118, 1015)-net in base 3, because

1 times m-reduction [i] would yield (69, 117, 1015)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66 613936 529260 501250 617603 459833 171908 716287 223417 644369 > 3¹¹⁷ [i]