MinT - Best Known (227−83, 227, s)-Nets in Base 3

Best Known (227−83, 227, s)-Nets in Base 3

(227−83, 227, 156)-Net over F₃ — Constructive and digital

Digital (144, 227, 156)-net over F₃, using

17 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 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]

(227−83, 227, 270)-Net over F₃ — Digital

Digital (144, 227, 270)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−83, 227, 3402)-Net in Base 3 — Upper bound on s

There is no (144, 227, 3403)-net in base 3, because

1 times m-reduction [i] would yield (144, 226, 3403)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 681997 298736 359630 748802 331899 801764 119113 702494 037532 127094 934728 553590 899257 675490 707777 955686 481887 042055 > 3²²⁶ [i]