MinT - Best Known (135−61, 135, s)-Nets in Base 3

Best Known (135−61, 135, s)-Nets in Base 3

(135−61, 135, 68)-Net over F₃ — Constructive and digital

Digital (74, 135, 68)-net over F₃, using

1 times m-reduction [i] based on digital (74, 136, 68)-net over F₃, using
- trace code for nets [i] based on digital (6, 68, 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]

(135−61, 135, 95)-Net over F₃ — Digital

Digital (74, 135, 95)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−61, 135, 785)-Net in Base 3 — Upper bound on s

There is no (74, 135, 786)-net in base 3, because

1 times m-reduction [i] would yield (74, 134, 786)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8769 385378 506060 799559 102682 124700 269343 331987 459598 811684 173965 > 3¹³⁴ [i]