MinT - Best Known (139−47, 139, s)-Nets in Base 3

Best Known (139−47, 139, s)-Nets in Base 3

(139−47, 139, 156)-Net over F₃ — Constructive and digital

Digital (92, 139, 156)-net over F₃, using

1 times m-reduction [i] based on digital (92, 140, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 70, 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]

(139−47, 139, 226)-Net over F₃ — Digital

Digital (92, 139, 226)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−47, 139, 3414)-Net in Base 3 — Upper bound on s

There is no (92, 139, 3415)-net in base 3, because

1 times m-reduction [i] would yield (92, 138, 3415)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 699196 707837 882005 955655 542216 141907 703662 657286 921857 499300 197299 > 3¹³⁸ [i]