MinT - Best Known (85, 85+57, s)-Nets in Base 3

Best Known (85, 85+57, s)-Nets in Base 3

(85, 85+57, 128)-Net over F₃ — Constructive and digital

Digital (85, 142, 128)-net over F₃, using

2 times m-reduction [i] based on digital (85, 144, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 72, 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, 85+57, 139)-Net over F₃ — Digital

Digital (85, 142, 139)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+57, 1400)-Net in Base 3 — Upper bound on s

There is no (85, 142, 1401)-net in base 3, because

1 times m-reduction [i] would yield (85, 141, 1401)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 031884 071915 515039 077937 638623 037274 448910 636057 711380 933337 196145 > 3¹⁴¹ [i]