MinT - Best Known (86−31, 86, s)-Nets in Base 3

Best Known (86−31, 86, s)-Nets in Base 3

(86−31, 86, 80)-Net over F₃ — Constructive and digital

Digital (55, 86, 80)-net over F₃, using

8 times m-reduction [i] based on digital (55, 94, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 47, 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]

(86−31, 86, 127)-Net over F₃ — Digital

Digital (55, 86, 127)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−31, 86, 1608)-Net in Base 3 — Upper bound on s

There is no (55, 86, 1609)-net in base 3, because

1 times m-reduction [i] would yield (55, 85, 1609)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35935 031926 378102 965154 663642 674311 848331 > 3⁸⁵ [i]