MinT - Best Known (85−33, 85, s)-Nets in Base 3

Best Known (85−33, 85, s)-Nets in Base 3

(85−33, 85, 80)-Net over F₃ — Constructive and digital

Digital (52, 85, 80)-net over F₃, using

3 times m-reduction [i] based on digital (52, 88, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 44, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(85−33, 85, 101)-Net over F₃ — Digital

Digital (52, 85, 101)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−33, 85, 1072)-Net in Base 3 — Upper bound on s

There is no (52, 85, 1073)-net in base 3, because

1 times m-reduction [i] would yield (52, 84, 1073)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12148 692160 387483 956246 868344 406758 529857 > 3⁸⁴ [i]