MinT - Best Known (42, 42+33, s)-Nets in Base 3

Best Known (42, 42+33, s)-Nets in Base 3

(42, 42+33, 60)-Net over F₃ — Constructive and digital

Digital (42, 75, 60)-net over F₃, using

1 times m-reduction [i] based on digital (42, 76, 60)-net over F₃, using
- trace code for nets [i] based on digital (4, 38, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]

(42, 42+33, 65)-Net over F₃ — Digital

Digital (42, 75, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+33, 531)-Net in Base 3 — Upper bound on s

There is no (42, 75, 532)-net in base 3, because

1 times m-reduction [i] would yield (42, 74, 532)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 203098 749306 139892 374931 178362 950785 > 3⁷⁴ [i]