MinT - Best Known (219−79, 219, s)-Nets in Base 3

Best Known (219−79, 219, s)-Nets in Base 3

(219−79, 219, 156)-Net over F₃ — Constructive and digital

Digital (140, 219, 156)-net over F₃, using

17 times m-reduction [i] based on digital (140, 236, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 118, 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]

(219−79, 219, 272)-Net over F₃ — Digital

Digital (140, 219, 272)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−79, 219, 3537)-Net in Base 3 — Upper bound on s

There is no (140, 219, 3538)-net in base 3, because

1 times m-reduction [i] would yield (140, 218, 3538)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 428663 610813 857526 876717 215577 902234 975500 514003 930255 752051 183853 861132 271548 049240 413321 305306 521065 > 3²¹⁸ [i]