MinT - Best Known (86−27, 86, s)-Nets in Base 3

Best Known (86−27, 86, s)-Nets in Base 3

(86−27, 86, 128)-Net over F₃ — Constructive and digital

Digital (59, 86, 128)-net over F₃, using

6 times m-reduction [i] based on digital (59, 92, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 46, 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]

(86−27, 86, 195)-Net over F₃ — Digital

Digital (59, 86, 195)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−27, 86, 3720)-Net in Base 3 — Upper bound on s

There is no (59, 86, 3721)-net in base 3, because

1 times m-reduction [i] would yield (59, 85, 3721)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36024 102290 149635 070152 342505 795228 395747 > 3⁸⁵ [i]