MinT - Best Known (59−27, 59, s)-Nets in Base 3

Best Known (59−27, 59, s)-Nets in Base 3

(59−27, 59, 40)-Net over F₃ — Constructive and digital

Digital (32, 59, 40)-net over F₃, using

1 times m-reduction [i] based on digital (32, 60, 40)-net over F₃, using
- trace code for nets [i] based on digital (2, 30, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]

(59−27, 59, 49)-Net over F₃ — Digital

Digital (32, 59, 49)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−27, 59, 368)-Net in Base 3 — Upper bound on s

There is no (32, 59, 369)-net in base 3, because

1 times m-reduction [i] would yield (32, 58, 369)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4742 841183 732198 775307 686835 > 3⁵⁸ [i]