MinT - Best Known (85−27, 85, s)-Nets in Base 3

Best Known (85−27, 85, s)-Nets in Base 3

(85−27, 85, 128)-Net over F₃ — Constructive and digital

Digital (58, 85, 128)-net over F₃, using

5 times m-reduction [i] based on digital (58, 90, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 45, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(85−27, 85, 186)-Net over F₃ — Digital

Digital (58, 85, 186)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−27, 85, 3417)-Net in Base 3 — Upper bound on s

There is no (58, 85, 3418)-net in base 3, because

1 times m-reduction [i] would yield (58, 84, 3418)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11987 393752 511164 272755 508941 269332 246717 > 3⁸⁴ [i]