MinT - Best Known (86, 86+39, s)-Nets in Base 3

Best Known (86, 86+39, s)-Nets in Base 3

(86, 86+39, 156)-Net over F₃ — Constructive and digital

Digital (86, 125, 156)-net over F₃, using

3 times m-reduction [i] based on digital (86, 128, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 64, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(86, 86+39, 264)-Net over F₃ — Digital

Digital (86, 125, 264)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+39, 5134)-Net in Base 3 — Upper bound on s

There is no (86, 125, 5135)-net in base 3, because

1 times m-reduction [i] would yield (86, 124, 5135)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 146007 176854 697664 575257 329710 948861 185753 720385 776689 456411 > 3¹²⁴ [i]