MinT - Best Known (54, 54+41, s)-Nets in Base 3

Best Known (54, 54+41, s)-Nets in Base 3

(54, 54+41, 68)-Net over F₃ — Constructive and digital

Digital (54, 95, 68)-net over F₃, using

1 times m-reduction [i] based on digital (54, 96, 68)-net over F₃, using
- trace code for nets [i] based on digital (6, 48, 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]

(54, 54+41, 82)-Net over F₃ — Digital

Digital (54, 95, 82)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+41, 706)-Net in Base 3 — Upper bound on s

There is no (54, 95, 707)-net in base 3, because

1 times m-reduction [i] would yield (54, 94, 707)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 718 795986 233356 004421 350257 183788 032600 937657 > 3⁹⁴ [i]