MinT - Best Known (234−89, 234, s)-Nets in Base 3

Best Known (234−89, 234, s)-Nets in Base 3

(234−89, 234, 156)-Net over F₃ — Constructive and digital

Digital (145, 234, 156)-net over F₃, using

12 times m-reduction [i] based on digital (145, 246, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 123, 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]

(234−89, 234, 247)-Net over F₃ — Digital

Digital (145, 234, 247)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−89, 234, 2857)-Net in Base 3 — Upper bound on s

There is no (145, 234, 2858)-net in base 3, because

1 times m-reduction [i] would yield (145, 233, 2858)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1493 947657 555163 625084 865313 921801 506920 285765 170666 048056 607109 807919 127535 952631 079225 445536 681336 886724 115129 > 3²³³ [i]