MinT - Best Known (50−19, 50, s)-Nets in Base 3

Best Known (50−19, 50, s)-Nets in Base 3

(50−19, 50, 68)-Net over F₃ — Constructive and digital

Digital (31, 50, 68)-net over F₃, using

trace code for nets [i] based on digital (6, 25, 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]

(50−19, 50, 74)-Net over F₃ — Digital

Digital (31, 50, 74)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−19, 50, 812)-Net in Base 3 — Upper bound on s

There is no (31, 50, 813)-net in base 3, because

1 times m-reduction [i] would yield (31, 49, 813)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 240325 704168 961731 556923 > 3⁴⁹ [i]