MinT - Best Known (218−84, 218, s)-Nets in Base 3

Best Known (218−84, 218, s)-Nets in Base 3

(218−84, 218, 156)-Net over F₃ — Constructive and digital

Digital (134, 218, 156)-net over F₃, using

6 times m-reduction [i] based on digital (134, 224, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 112, 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]

(218−84, 218, 224)-Net over F₃ — Digital

Digital (134, 218, 224)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−84, 218, 2431)-Net in Base 3 — Upper bound on s

There is no (134, 218, 2432)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 933768 553347 259791 071711 275714 157720 636837 513807 813092 632942 264747 314083 105444 000242 487642 167181 250817 > 3²¹⁸ [i]