MinT - Best Known (219−83, 219, s)-Nets in Base 3

Best Known (219−83, 219, s)-Nets in Base 3

(219−83, 219, 156)-Net over F₃ — Constructive and digital

Digital (136, 219, 156)-net over F₃, using

9 times m-reduction [i] based on digital (136, 228, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 114, 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]

(219−83, 219, 236)-Net over F₃ — Digital

Digital (136, 219, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−83, 219, 2737)-Net in Base 3 — Upper bound on s

There is no (136, 219, 2738)-net in base 3, because

1 times m-reduction [i] would yield (136, 218, 2738)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 910643 288667 994709 684702 048257 118187 722796 864654 465348 697460 059195 640848 912135 721252 345954 229732 065877 > 3²¹⁸ [i]