MinT - Best Known (227−87, 227, s)-Nets in Base 3

Best Known (227−87, 227, s)-Nets in Base 3

(227−87, 227, 156)-Net over F₃ — Constructive and digital

Digital (140, 227, 156)-net over F₃, using

9 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]

(227−87, 227, 236)-Net over F₃ — Digital

Digital (140, 227, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−87, 227, 2674)-Net in Base 3 — Upper bound on s

There is no (140, 227, 2675)-net in base 3, because

1 times m-reduction [i] would yield (140, 226, 2675)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 676929 197183 292011 756916 369700 649798 806417 172113 929553 898038 204302 300216 954142 157310 231705 445690 327819 840155 > 3²²⁶ [i]