MinT - Best Known (51−19, 51, s)-Nets in Base 3

Best Known (51−19, 51, s)-Nets in Base 3

(51−19, 51, 68)-Net over F₃ — Constructive and digital

Digital (32, 51, 68)-net over F₃, using

1 times m-reduction [i] based on digital (32, 52, 68)-net over F₃, using
- trace code for nets [i] based on digital (6, 26, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]

(51−19, 51, 80)-Net over F₃ — Digital

Digital (32, 51, 80)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−19, 51, 919)-Net in Base 3 — Upper bound on s

There is no (32, 51, 920)-net in base 3, because

1 times m-reduction [i] would yield (32, 50, 920)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 723447 764334 258366 742449 > 3⁵⁰ [i]