MinT - Best Known (227−77, 227, s)-Nets in Base 3

Best Known (227−77, 227, s)-Nets in Base 3

(227−77, 227, 162)-Net over F₃ — Constructive and digital

Digital (150, 227, 162)-net over F₃, using

9 times m-reduction [i] based on digital (150, 236, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 118, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(227−77, 227, 337)-Net over F₃ — Digital

Digital (150, 227, 337)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−77, 227, 5131)-Net in Base 3 — Upper bound on s

There is no (150, 227, 5132)-net in base 3, because

1 times m-reduction [i] would yield (150, 226, 5132)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 678113 873012 157366 166117 195892 817251 077037 348104 300332 312698 052931 648046 168822 224751 421053 373894 492557 251593 > 3²²⁶ [i]